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THE  ACCOUNTANCY  OF 
INVESTMENT 


BY 

Charles  Ezra  Sprague,  a.m.,  Ph.d.,  Litt.d.,  c.p.a. 

Late  Professor  of  Accountancy  in  New   York    University ; 
Late  President  of  Union  Dime  Savings  Bank,  New  York  Ctty 


With  which  are  incorporated  "Logarithms  to  12  Places 

and  Their  Use  in   Interest  Calculations"   and 

"Amortization"  by  the  same  author 


REVISED  BY 

LEROY  L.  PeRRINE,  Ph.B.,  B.C.S.,  C.P.A. 

Lecturer  on  Accounting  at  the  Nexv  York  University  School 
of  Commerce,  Accounts  and  Finance;  Meynber  of  the 
staff  of  Haskins  &>  Sells,  Certified  Public  AccountaMis 


RONALD  ACCOUNTING   SERIES 


NEW  YORK 

THE  RONALD  PRESS  COMPANY 
1914 


Copyright,  1904,  by  Charles  E.  Sprague 


Copyright,  1905,  by  Charles  E.  Sprague 


Copyright,  1907,  by  Charles  E.  Sprague 


Copyright,  1909,  by  Charles  E.  Sprague 


Copyright,  1910,  by  Charles  E.  Sprague 


Copyright,  1914, 

by 

THE  RONALD  PRESS  COMPANY 


(Sjenjeral  SMtmmt  oi  ^laxiaxml  §0arir 


Applicable  to  all  books  of  the 
Ronald  Accounting  Series 


THE  manuscripts  of  the  books  forming  the 
Ronald  Accounting  Series  have  been  sub- 
mitted to  us  and  have  been  approved  by  us  for 
publication. 

In  some  cases  the  authors  express  views  that 
are  not  fully  in  accord  with  those  entertained  by 
us,  but  in  no  instance  are  such  differences  of 
sufficient  importance,  in  our  judgment,  to  warrant 
the  withholding  from  publication  of  a  meritorious 
work. 

J.  E.  Sterrett 

Robert  H.  Montgomery 


PREFACE 

Among  the  published  works  of  the  late  Colonel  Sprague, 
there  are  four  which  deal  particularly  with  certain  mathe- 
matical phases  of  accounting,  viz. :  *Text  Book  of  the  Ac- 
countancy of  Investment" ;  "Amortization" ;  "Logarithms  to 
12  Places  and  Their  Use  in  Interest  Calculations";  and 
"Extended  Bond  Tables."  Since  the  author's  death  in 
March,  1912,  it  has  become  desirable  to  combine  the  first 
three  of  these  publications  into  one  volume,  in  order  to 
serve  more  effectively  the  needs  of  business  men  and  students 
of  accounting  by  presenting  the  material  in  compact  and 
convenient  form. 

The  present  volume  is  the  result  of  this  consolidation.  In 
it  has  been  incorporated  everything  of  practical  value  con- 
tained in  the  three  works  mentioned,  while  at  the  same  time 
the  special  features  of  those  books  have  been  amplified  by 
additional  text  matter  and  problems,  wherever  such  addi- 
tions have  seemed  desirable  for  the  sake  of  more  adequate 
treatment. 

In  conformity  with  the  usual  and  commendable  practice 
of  Colonel  Sprague,  the  reviser  has  avoided  as  far  as  pos- 
sible the  use  of  the  more  difficult  mathematical  demonstra- 
tions and  formulas,  believing  that  thereby  the  book  will 
prove  of  greater  utility  to  practicing  accountants,  bankers, 
and  other  business  men.  On  this  point  we  quote  from  the 
author's  original  preface :  "Treatises  on  this  subject  (Mathe- 
matics of  Investment),  written  for  actuarial  students,  are 
invariably  too  difficult,  except  for  those  who  have  not  only 
been  highly  trained  in  algebra,  but  are  fresh  in  its  use,  and 
this  makes  the  subject  forbidding  to  many  minds.     I  have 


vi  PREFACE 

made  all  my  demonstrations  arithmetical  and  illustrative, 
but,  I  think,  none  the  less  convincing  and  intelligible." 

It  is  believed  that  for  a  considerable  number  of  readers, 
the  tables  of  logarithms  given  in  Part  III  will  prove  of  great 
utility  in  those  cases  where  more  than  ordinary  accuracy  is 
required,  and  where  special  tables  are,  at  times,  imperative. 
To  quote  again  from  the  preface  of  Colonel  Sprague: 
"Rough  results  will  answer  for  approximative  purposes ;  but 
where  it  is  desirable,  for  instance,  to  construct  a  table  of 
amortization,  sinking  fund,  or  valuation  of  a  lease  at  an 
unusual  rate,  for  a  large  amount  and  for  a  great  many  years, 
exactness  is  desirable  and  becomes  self-proving  at  the  end." 

A  whole  book  is  required  for  the  ordinary  tables  of  loga- 
rithms of  six  or  seven  places,  while  the  tables  here  presented 
are  contained  in  a  few  pages  and  give  accurate  results  to 
twelve  places  of  decimals.  The  processes  with  these  tables 
are  necessarily  somewhat  slower  than  with  those  of  six  or 
seven  places,  but  their  use  is  fully  justified  where  greater 
accuracy  in  results  is  desirable. 

The  tables  of  compound  interest,  present  worth,  an- 
nuities, and  sinking  funds,  carried  to  eight  places  of  decimals, 
have  been  retained  in  this  edition.  Such  tables  are  prac- 
tically indispensable  in  securing  accurate  computations.  The 
index  of  subjects  at  the  end  of  the  book  is  a  new  feature  and 
will  facilitate  quick  reference  to  any  information  desired. 

Leroy  L.  Perrine. 
New  York  City,  January,  1914. 


CONTENTS 


PART  I— THE  MATHEMATICS  OF  INVESTMENT 

Chapter  I.    Capital  and  Revenue 

•  1.  Definition  of  Capital 

2.  The  Use  of  Capital 

3.  Sources  of  Capital 

4.  Investment  ^ 

5.  Revenue 

6.  Interest  and  Rent 

7.  Dividends 

8.  Laws  of  Interest 

Chapter  II.    Interest 

I  9.  Interest 

10.  Essentials  of  Interest  Contract 

11.  Interest  Rate 

12.  Principal 

13.  Simple  and  Compound  Interest 

14.  Punctual  Interest 

15.  Computation  of  Interest 

16.  Simple  Interest 

17.  Compound  Interest 

18.  Computation  of  Compound  Interest 

19.  Comparison  of  Simple  and  Compound  Interest 

20.  The  Day  as  a  Time  Unit 

21.  The  Month  as  a, Time  Unit 

22.  Half  and  Quarter  Years 

23.  Partial  Interest  Periods 

24.  Changing  the  Day  Basis 

25.  The  Amount — First  Period 

26.  The  Amount — Subsequent  Periods 

27.  Exponents  and  Powers 

28.  Finding  the  Amount — Compound  Interest 

29.  Present  Worth 

30.  Present  Worth  and  Amount  Series 

31.  Relation  Between  Present  Worth  and  Amount 


iii  CONTENTS 

32.  Formation  of  Series 

33.  Discount 

34.  Computing  Compound  Discount 

35.  Formulas  for  Interest  Calculations 

36.  Use  of  Logarithms 

Chapter  III.    The  Use  of  Logarithms 

§  37.  Purpose  of  Logarithms 

38.  Exponents,  Powers,  and  Roots 

39.  Logarithms  as  Exponents 

40.  Rules  and  Symbols  of  Logarithms  ^ 

41.  The  Two  Parts  of  a  Logarithm 

42.  Mantissa  Not  Affected  by  Position  of  the  Decimal  Point 

43.  Four-Place  Table  of  Logarithms 

44.  Multiplication  by  Logarithms 

45.  Division  by  Logarithms 

46.  Powers  by  Logarithms 

47.  Roots  by  Logarithms 

48.  Fractional  Exponents 

49.  Use  of  Logarithms  in  Computing  Compound  Interest 

50.  Accuracy  of  Logarithmic  Results 

51.  Logarithms  to  Fifteen  Places 

52.  Use  of  Logarithms  in  Present  Worth  Calculations 

Chapter  IV.    Amount  of  an  Annuity 

§  53.  Evaluation  of  a  Series  of  Payments 

54.  Annuities 

55.  Amount  of  Annuity 

56.  Calculation  of  Annuity  Amounts 

57.  Formation  of  Tables 

58.  Use  of  Tables 

59.  Compound  Interest  as  a  Base  for  Annuity  Calculations 

60.  Rule  and  Formula  for  Finding  Amount 

61.  Operation  of  Rule 

Chapter  V.    Present  Worth  of  an  Annuity 

§  62.  Method  of  Calculation 

63.  Tables  of  Present  Worth 

64.  Short  Method  for  Finding  Present  Worth  of  an  Annuity 

65.  Present  Worth  Obtained 

66.  Rule  for  Present  Worth 

67.  Formulas  for  Present  Worth 

68.  Analysis  of  Annuity  Payments 

69.  Components  of  Annuity  Instalments 


CONTENTS  ix 

70.  Amortization 

71.  Amortization  and  Present  Worth 

72.  Development  of  a  Series  of  Amortizations 
7Z.  Evaluation  by  Logarithms 

Chapter  VI.    Special  Forms  of  Annuities 

74.  Ordinary  or  Immediate  Annuities 

75.  Annuities  Due 

76.  Present  Worth  of  Annuities  Due 

77.  Present  Worth  of  Deferred  Annuities 

78.  Rule  for  Finding  Present  Worth  of  Deferred  Annuity 

79.  Present  Worth  of  Perpetuities 

80.  Perpetuity  in  Stock  Purchased  for  Investment 

81.  When  Annuity  Periods  and  Interest  Periods  Differ 

82.  Varying  Annuities 

Chapter  VII.    Rent  of  Annuity  and  Sinking  Fund 

83.  Rent  of  Annuity 

84.  Rule  for  Finding  Rent  of  Annuity 

85.  Alternative  Method  of  Finding  Rent 

86.  Rent  of  Deferred  Payments 

87.  Annuities  as  Sinking  Funds 

88.  Rule  for  Finding  Sinking  Fund  Contributions 

89.  Verification  Schedule 

90.  Amortization  and  Sinking  Fund 

Chapter  VIII.    Nominal  and  Effective  Rates 

91.  Explanation  of  Terms 

92.  Semi-Annual  and  Quarterly  Conversions 

93.  Limit  of  EflFective  Annual  Rate 

94.  Rule  for  EflFective  Rate 

95.  Logarithmic  Process 

Chapter  IX.    Bonds  and  the  Proper  Basis  of  Bond  Accounts 

I  96.  Provisions  of  Bonds 

97.  Interest  on  Bonds 

98.  Hov^r  Bonds  Are  Designated 

99.  Relation  of  Cost  to  Net  Income 

100.  Coupon  and  EflFective  Rate  of  Interest  on  Bonds 

101.  Present  Worth  of  Bonds 

102.  Considerations  in  the  Purchase  of  Bonds 

103.  Present  Worth  and  Earning  Capacity  of  Bonds 

104.  Cost  and  Par  of  Bonds 


CONTENTS 

105.  Intermediate  Value  of  Bonds 

106.  True  Investment  Basis  for  Bonds 

107.  Various  Bond  Values 

108.  Investment  Value  the  True  Accounting  Basis 

Chapter  X.    Valuation  of  Bonds 

109.  Cash  Rate  and  Income  Rate  of  Bonds 

110.  Elements  of  a  Bond 

111.  Valuation  of  Bonds — First  Method 

112.  (a)  Finding  Present  Worth  of  Principal 

113.  (b)  Present  Worth  of  Coupons 

114.  Schedule  of  Evaluation 

115.  Valuation  of  Bonds — Second  Method 

116.  Evaluation  When  Cash  Rate  Is  Less  Than  Income  Rate 

117.  Second  Method  by  Schedule 

118.  Rule  for  Second  Method  of  Evaluation 

119.  Principles  of  Investment 

120.  Solution  by  Logarithms 

121.  Amortization  Schedule 

122.  Use  of  Schedules  in  Accountancy 

123.  Book  Values  in  Schedules 

124.  Checks  on  Accuracy  of  Schedules 

125.  Tables  Derivable  from  Bond  Values 

126.  Methods  of  Handling  Interest 

127.  Schedule  of  Bond  Values 

Chapter  XI.    Valuation  of  Bonds  (Concluded) 

128.  Bond  Purchases  at  Intermediate  Dates 

129.  Errors  in  Adjusting  Bond  Prices 

130.  Schedule  of  Periodic  Evaluation 

131.  Objection  to  Valuation  on  Interest  Dates 

132.  Interpolation  Method  of  Periodic  Valuation 

133.  Multiplication  Method  of  Valuation 

134.  Computation  of  Net  Income  for  Partial  Period 

135.  Purchase  Agreements 

136.  Approximation  Method  of  Finding  Income  Rate 

137.  Application  of  Method 

138.  First  Method  of  Eliminating  Residues 

139.  Second  Method  of  Eliminating  Residues 

140.  Third  Method  of  Eliminating  Residues 

141.  Short  Terminals 

142.  Rule  for  Short  Terminals 

143.  Discounting 

144.  Last  Half- Year  of  Bond 


CONTENTS  ,  xi 

145.  Serial  Bonds 

146.  Irredeemable  Bonds 

147.  Optional  Redemption 

148.  Bonds  as  Trust  Fund  Investments 

149.  Payments  to  Life  Tenant 

150.  Effect  of  Varying  Rates  on  Investments 

151.  Example  of  Payments  to  Life  Tenant 

152.  Cullen  Decision 

153.  Cullen  Decision  Scheduled 

154.  Unjust  Feature  of  Cullen  Decision 

155.  Bond  Tables 

156.  Features  of  the  Bond  Table 

Chapter  XII.    Summary  of  Compound  Interest  Processes 

§  157.  Rules  and  Formulas 

158.  Rules 

159.  Formulas 

Chapter  XIII.    Accounts — General  Principles 

§  160.  Relation  of  General  Ledger  to  Subordinate  Ledgers 

161.  The  Interest  Account 

162.  Mortgage  and  Loan  Accounts 

Chapter  XIV.    Real  Estate  Mortgages 

§  163.  Nature  of  Loans  on  Bond  and  Mortgage 

164.  Separate  Accounts  for  Principal  and  Interest 

165.  Interest  Debits  and  Credits 

166.  Characteristics  of  Modern  Ledger 

167.  The  Mortgage  Ledger 

168.  Identification  of  Mortgages  by  Number 

169.  The  "Principal"  Account 

170.  Special  Columns  for  Mortgagee's  Disbursements 

171.  The  Interest  Account 

172.  Interest  Due 

173.  Books  Auxiliary  to  Ledger 

174.  The  "Due"  Column 

175.  Interest  Account  Must  Be  Analyzed 

176.  Form  of  "Interest  Due"  Account 

177.  Forms  for  Mortgage  Account 

178.  Loose-Leaf  and  Card  Records 

179.  Forms  of  Mortgage  Loan  Accounts 

180.  Reverse  Posting  of  Interest  Register 

181.  Handling  Receipts  and  Notices 

182.  Mortgages  Account  in  General  Ledger 


ii  CONTENTS 

183.  Tabular  Register 

184.  Equal  Instalment  Method 

Chapter  XV.    Loans  on  Collateral 

185.  Short  Time  Loans  on  Personal  Property 

186.  Forms  for  Loan  Accounts 

187.  Requirements  for  Interest  Account 

188.  Forms  for  Collateral  Loan  Accounts 

Chapter  XVI.    Interest  Accounts 

189.  Functions  of  the  Three  Interest  Accounts 

190.  A  Double  Record  for  Interest  Earned 

191.  Example  of  Interest  Income 

192.  Daily  Register  of  Interest  Accruing 

193.  Monthly  Summary 

194.  Method  and  Importance  of  Interest  Earned  Account 

195.  Interest  Accounts  in  General  Ledger 

196.  Payment  of  Accrued  Interest 

Chapter  XVII.    Bonds  and  Similar  Securities 

197.  Investments  with  Fluctuating  Values 

198.  Amortization  Account 

199.  Effect  on  Schedule  of  Additional  Purchases 

200.  The  Bond  Sales  Account 

201.  Requirements  as  to  Bond  Records 

202.  Form  of  Bond  Ledger 

203.  Interest  Due  Account 

204.  Interest  Account — Bond  Ledger 

205.  Amortization  Entries 

206.  Bond  Entries  in  General  Ledger 

207.  Accounts  Where  Original  Cost  Is  Disregarded 

208.  Amortization  Reserve 

209.  Premiums  and  Amortization 

210.  Writing  Off  Premiums 

211.  Disposal  of  Amortization 

212.  Amortization  Accounting — Comparison  of  Methods 

213.  Irredeemable  Bonds  a  Perpetual  Annuity 

214.  Bond  Accounts  for  General  Ledger 

(Plans  I  to  V) 

Chapter  XVIII.    Discounted  Values 

215.  Securities  Payable  at  Fixed  Dates  Without  Interest 

216.  Rates  of  Interest  and  Discount 

217.  Rate  of  Discount  Named  in  Notes 


CONTENTS  xiii 


218.  Form  as  AfiFecting  Legality 

219.  Entry  of  Notes  Discounted 

220.  Discount  and  Interest  Entries 

221.  Total  Earnings  from  Discounts 


PART  II— PROBLEMS  AND  STUDIES 
Chapter  XIX.    Interest  and  Discount 

§  222.  Problems  in  Simple  Interest 

223.  Notes  on  the  One  Per  Cent  Method 

224.  Answers  to  Problems  in  Simple  Interest 

225.  Problems  in  Compound  Interest 

226.  Answers  to  Problems  in  Compound  Interest 

227.  Proof  of  Amount  and  Present  Worth 

228.  Contracted  Multiplication 

229.  Problems  in  the  Use  of  Logarithms 

230.  Problems    Requiring    Use    of    More    Extended   Tables    of 

Logarithms 

231.  Answers  to  Problems  in  Logarithms 

Chapter  XX.    Problems  in  Annuities  and  in  Nominal  and  Effective 

Rates 

§  232.  Problems  in  Annuities 

233.  Answers  to  Problems  in  Annuities 

234.  Problems  in  Rent  of  Annuity  and  Sinking  Fund 

235.  Answers  to  Problems  in  Rent  of  Annuity  and  Sinking  Fund 

236.  Problems  in  Nominal  and  Effective  Rates 

237.  Answers  to  Problems  in  Nominal  and  Effective  Rates 

238.  Constant  Compounding 

239.  Finding  Nominal  Rate 

240.  Approximate  Rules 

Chapter  XXI.    Equivalent  Rates  of  Interest— Bond  Valuations 

§  241.  Annual  and  Semi-Annual  Interest 

242.  Semi-Annual  and  Quarterly  Interest 

243.  Problems  in  Valuation  of  Bonds 

244.  Successive  Method  of  Bond  Valuation — Problems 

245.  Answers  to  Bond  Valuation  Problems 

246.  Bond  Valuations  by  the  Use  of  Logarithms 

247.  Finding  Initial  Book  Values 

248.  Tabular  Multiplication  and  Contracted  Division 

249.  Formation  of  Successive  Amortizations 


xiv  CONTENTS 

250.  Test  by  Differencing 

251.  Successive  Columns 

252.  Intentional  Errors 

253.  Rejected  Decimals 

254.  Limit  of  Tolerance 

Chapter  XXII.     Broken  Initial  and  Short  Terminal  Bonds 

§  255.  Problems  in  Valuation 

256.  Answers  to  Valuation  Problems 

Chapter  XXIII.    The  Use  of  Tables  in  Determining  the  Accurate 

Income  Rate 

§  257.  Bond  Tables  as  Annuity  Tables 

258.  Premium  and  Discount  as  a  Present  Worth 

259.  Present  Worth  by  Differences 

260.  Present  Worth  by  Division 

261.  Compound  Discount  and  Present  Value  of  a  Single  Sum 

262.  Use  of  Bond  Tables  in  Compound  Interest  Problems 

263.  Determination  of  Accurate  Income  Rate 

264.  Assumed  Trial  Rate 

265.  Application  of  Assumed  Trial  Rate — Bond  Above  Par 

266.  Variations  in  Assumed  Rates 

267.  Application  of  Assumed  Trial  Rate — Bond  Below  Par 

268.  Trial  Rates  from  Bond  Tables 

269.  Use  of  Bond  Tables 

Chapter  XXIV.     Discounting 

§  270.  Table  of  Multiples 

271.  Present  Worths  of  Interest-Difference 

272.  Discounts  from  Tables 

273.  Reussner's  Tables 

Chapter  XXV.    Serial  Bonds 

§  274.  Problem  in  Valuation  of  Serial  Bonds 

275.  Inter-rates 

276.  Table  of  Differences 

277.  Successive  Method 

278.  Balancing  Period 

279.  First  Payment  in  Series 

280.  Elimination  of  Residue 

281.  Schedule 

282.  Uneven  Loans 

283.  Tabular  Methods 


CONTENTS  XV 

284.  Formula  for  Serials 

285.  Problems  in  Valuation  of  Serial  Bonds 

286.  Answers  to  Problems  in  Valuation  of  Serial  Bonds 

Chapter  XXVI.    Option  of  Redemption 

287.  Method  of  Calculating  Income  Rate 

288.  Advantageous  Redemption  Ignored 

289.  Disadvantageous  Redemption  Expected 

290.  Change  in  Principal 

291.  Approximate  Location 

292.  Problems  Involving  Optional  Redemption  Dates 

293.  Rule  for  Determining  Net  Income 

294.  Answers  to  Problems  Involving  Optional  Redemption  Dates 

Chapter  XXVII.     Bonds  at  Annual  and  Other  Rates 

295.  Standard  of  Interest 

296.  Semi-Annual  and  Quarterly  Coupons 

297.  Shifting  of  Income  Basis 

298.  Problems — Bonds  at  Varying  Rates 

299.  Answers  to  Problems — Bonds  at  Varying  Rates 

300.  Bonds  with  Annual  Interest — Semi-Annual  Basis 

301.  Annualization 

302.  ,  Semi-Annual  Income  Annualized 

303.  Comparison  of  Annual  and  Semi-Annual  Bonds 

304.  Finding  Present  Worth  of  an  Annuity 

305.  Rule  for  Bond  Valuation 

306.  Multipliers  for  Annualizing 

307.  Formula  for  Annualizer 

308.  Conventional  Process 

309.  Scientific  Process 

310.  Values  Derived  from  Tables 

311.  Successive  Process 

312.  Problems  and  Answers — Varying  Time  Basis 

313.  Bonds  at  Two  Successive  Rates 

314.  Calculation  of  Immediate  Premium 

315.  Calculation  of  Deferred  Premium 

316.  Symbols  and  Rule 

317.  Analysis  of  Premiums 

318.  Problems  and  Answers — Successive  Rates 

Chapter  XXVIII.     Repayment  and  Reinvestment 

319.  Aspects  of  Periodic  Payment 

320.  Integration  of  Original  Debt 

321.  Use  of  the  Reinvestment  Point  of  View 

322.  Replacement 


xvi  CONTENTS 

323.  Diminishing  Interest  Rates 

324.  Proof  of  Accuracy 

325.  Varying  Rates  of  Interest 

326.  Dual  Rate  for  Income  and  Accumulation 

327.  Instalment  at  Two  Rates 

328.  Amortization  of  Premiums  at  Dual  Rate 

329.  Modified  Method  for  Valuing  Premiums 

330.  Rule  for  Valuation  of  a  Premium 

331.  Computation  at  Dual  Rate 

332.  Dual  Rate  in  Bookkeeping 

333.  Utilization  of  Dual  Principle 

334.  Installation  of  Amortization  Accounts 

335.  Scope  of  Calculations 

336.  Method  of  Procedure  When  Same  Basis  Is  Retained 


PART  III— LOGARITHMS 
Chapter  XXIX.    Finding  a  Number  When  Its  Logarithm  Is  Given 

§  2Z7.  Logarithmic  Tables 

338.  Discussion  of  Logarithms 

339.  Standard  Tables  of  Logarithms 

340.  United  States  Coast  Survey  Tables 
34L  Gray  and  Steinhauser  Tables 

342.  A  Twelve-Place  Table 

343.  The  "Factoring"  Method 

344.  Finding  a  Number  from  Its  Logarithm 

345.  Procedure  in  an  Unusual  Case 

346.  Rule  for  Finding  Number  When  Logarithm  Is  Given 

347.  Method  by  Multiples 

Chapter  XXX.    Forming  Logarithms;  Tables 

§  348.  Explanation  of  Process 

349.  Rule  for  Finding  a  Logarithm 

350.  Examples  of  Logarithmic  Computations 

351.  Logarithms  to  Less  Than  Twelve  Places 

352.  Tables  with  More  Than  Twelve  Places 

353.  Multiplying  Up 

354.  Process  of  Multiplying  Up 

355.  Supplementary  Multiplication 

356.  Multiplying  Up  by  Little's  Table 

357.  Different  Bases 

358.  Table  of  Factors 

359.  Table  of  Interest  Ratios 


CONTENTS  xvii 


360.  Table  of  Sub-Reciprocals 

361.  Table  of  Multiples 


PART  IV— TABLES 
Chapter  XXXI.    Explanation  of  Tables  Used 

§  362.  Object  of  the  Tables 

363.  Degree  of  Accuracy 

364.  Rates  and  Periods 

365.  Tables  Shown 

366.  Annuities — When  Payable 

367.  Table  I— Amount 

368.  Compound  Interest 

369.  Table  II— Present  Worth 

370.  Table  III— Amount  of  Annuity 

371.  Table  IV— Present  Worth  of  Annuity 

372.  Table  V— Sinking  Fund 

373.  Rent  of  Annuity 

374.  Extension  of  Time 

375.  Subdivision  of  Rates 

376.  Interpolation 

377.  Table  VI — Reciprocals  and  Square  Roots 

Chapter  XXXII.    Tables  of  Compound  Interest,  Present  Worth, 
Annuities,  Sinking  Funds,  and  Other  Computations 

§  378.    Table  I — Amount  of  $1  at  Compound  Interest 

(a)  Part  1 

(b)  Part  2 

379.  Table  II— Present  Worth  of  $1  at  Compound  Interest 

(a)  Part  1 

(b)  Part  2 

380.  Table  III— Amount  of  Annuity  of  $1  at  End  of  Each  Period 

(a)  Part  1 

(b)  Part  2 

381.  Table  IV— Present  Worth  of  Annuity  of  $1  at  End  of  Each 

Period 

(a)  Part  1 

(b)  Part  2 

382.  Table  V — Sinking  Fund  or  Annuity  Which,  Invested  at  the 

End  of  Each  Period,  Will  Amount  to  $1 

(a)  Part  1 

(b)  Part  2 

383.  Table  VI — Reciprocals  and  Square  Roots 


EXPLANATION  OF  SYMBOLS 


For  the  sake  of  brevity  and  clearness,  certain  constantly- 
recurring  expressions  have  been  represented  in  the  text  by 
symbols.  The  following  list  comprises  all  of  those  which  are 
not  self-explanatory. 

1  =  $1,  £l,  or  any  other  unit  of  value. 

a  =  the  amount  of  $1  for  a  given  time  at  a  given  rate. 

A  =  the  amount  of  an  annuity  of  $1  for  a  given  time  at  a 
given  rate. 

c  =  the  cash,  or  coupon,  rate  of  interest  (or  the  cash  pay- 
ment) for  a  single  period. 

d  =  the  rate  of  discount  for  a  single  period. 

D  =  the  discount  on  $1  for  a  given  time  at  a  given  rate. 

i   =  the  rate  of  interest  (or  the  income)  for  a  single  period. 

I  =the  compound  interest  on  $1  for  a  given  time  at  a 
given  rate.. 

/  =  the  effective  rate  of  interest  for  one  year. 

M  =  an  indefinite  number  of  units. 

p  =  the  present  worth  of  $1  for  a  given  time  at  a  given  rate. 

P  =  the  present  worth  of  an  annuity  of  $1  for  a  given  time 
at  a  given  rate. 

r  =  (1+ 1),  the  periodic  ratio  of  increase. 


xvm 


THE  ACCOUNTANCY  OF  INVESTMENT 
Part  I — The  Mathematics  of  Investment 


CHAPTER  I 

CAPITAL   AND    REVENUE 

§  I.     Definition    of  Capital 

That  portion  of  wealth  which  is  set  aside  for  the  pro- 
duction of  additional  wealth  is  capital.  The  capital  of  a 
business,  therefore,  is  the  whole  or  a  part  of  the  assets  of 
the  business,  and  of  course  appears  on  the  active  or  debit 
side  of  its  balance  sheet.  This  is  the  sense  in  which  the 
word  "capital"  is  used  in  economics ;  but  in  bookkeeping  the 
term  "Capital  account"  is  often  used  in  quite  another  sense 
to  mean  accounts  on  the  credit  or  passive  side,  which  denote 
proprietorship.  To  prevent  confusion,  the  use  of  the  ex- 
pression "Capital  account"  will  be  avoided. 

§  2.    The  Use  of  Capital 

In  active  business,  capital  must  be  employed,  and,  in 
order  to  produce  more  wealth,  it  must  be  combined  with 
skill  and  industry.  Businesses,  and  consequently  their  ac- 
counting methods,  vary  as  to  the  manner  in  which  capital 
is  used.  Cash  is  convertible  into  potential  capital  of  any 
kind  desired.    In  a  manufacturing  business  it  is  exchanged 

^9 


"20'  ^''  '  ^'^^riE^MAXriEMATICS  OF  INVESTMENT 

for  machinery,  appliances,  raw  materials,  and  labor  which 
transforms  these  raw  materials  into  finished  products.  In 
a  mercantile  business  cash  is  expended  for  goods,  bought  at 
one  price  to  sell  at  another,  and  for  collecting,  displaying, 
caring  for,  advertising,  and  delivering  goods.  To  bridge 
over  the  time  between  selling  and  collecting,  additional 
capital  is  required,  usually  known  as  "working  capital,"  but 
which  might  more  appropriately  be  styled  "waiting  capital." 
Thus  we  may  analyze  each  kind  of  business,  and  show  that 
the  nature  of  its  capital  assets  depends  on  the  character  of 
the  business. 

§  3.    Sources  of  Capital 

On  the  credit  side  of  the  balance  sheet  the  capital  must 
be  accounted  for  in  such  a  manner  as  to  show  its  sources. 
Here  there  are  two  sharply  divided  classes :  loan-csipitsl,  or 
liability,  and  ow^-capital,  or  proprietorship.  The  great 
distinction  is  that  the  latter  participates  in  the  profits  and 
bears  the  losses,  while  the  former  takes  its  share  irrespective 
of  the  success  of  the  concern.  It  is  the  own-capital  which 
is  referred  to  in  the  phrase  "Capital  account." 

§  4.    Investment 

While  we  often  speak  of  a  man's  capital  as  being  in- 
vested in  a  business,  we  use  the  word  "investment"  more 
strictly  when  we  confine  it  to  the  non-participating  sense. 
Thus  we  say,  "He  not  only  owns  a  business,  but  he  has 
some  investments  besides."  In  the  strictest  sense,  then,  in- 
vestment implies  divesting  one's  self  of  the  possession  and 
control  of  one's  assets,  and  granting  such  possession  and 
control  to  another.  The  advantage  of  the  use  of  capital 
must  be  great  enough  to  enable  the  user  to  earn  more  than 
the  sum  which  he  pays  to  the  investor,  or  capitalist.  There 
are  many  cases  where  the  surrender  is  not  absolute,  and 


CAPITAL  AND  REVENUE  21 

more  or  less  risk  is  assumed  by  the  investor.  This  is  not 
absolute  investment,  but  to  some  extent  partnership.  The 
essence  of  strict  investment  is  the  vicarious  earning  of  a 
share  in  gains  which  do  not  depend  on  the  business  skill  of 
the  investor. 

§  5.    Revenue 

All  investments  are  made  with  a  view  to  obtaining 
revenue,  which  is  the  share  of  the  earnings  given  for  the 
use  of  capital.  Revenue  takes  three  forms:  interest,  rent, 
and  dividends — the  first  two  corresponding  to  strict  invest- 
ment, and  the  latter  to  participation. 

§6.    Interest  and  Rent 

These  do  not  essentially  differ.  Both  are  stipulated  pay- 
ments for  the  use  of  capital ;  but  in  case  of  rent  the  identical 
physical  asset  received  by  the  lessee  must  be  returned  by 
him  on  the  completion  of  the  contract.  If  you  borrow  a 
dollar,  you  may  repay  any  dollar  you  please ;  if  you  hire  a 
house  or  a  horse,  you  may  not  return  any  house  or  any 
horse,  but  must  produce  the  identical  one  you  had.  Interest 
and  rent  are  both  proportionate  to  time. 

§  7.     Dividends 

These  are  profits  paid  over  to  the  owners  of  the  own- 
capital,  whether  partners  or  shareholders.  The  amount  is 
subtracted  from  the  collective  assets  and  paid  over  to  the 
separate  owners.  Theoretically  there  is  neither  profit  nor 
loss  in  this  distribution.  I  have  more  cash,  but  my  share 
in  the  collective  assets  is  exactly  that  much  less.  The  cash 
is  distributed  partly  because  it  is  needed  by  the  participants 
for  consumption ;  and  partly  because  no  more  capital  can  be 
profitably  used  in  the  enterprise.  Some  concerns,  however, 
such  as  banks,  which  can  profitably  use  more  capital  and 


22  THE  MATHEMATICS  OF  INVESTMENT 

whose  shareholders  do  not  require  cash  for  consumption, 
frequently  refrain  from  dividing  the  periodical  profits,  or 
distribute  but  a  small  portion  of  them. 

The  accumulation  of  the  profits,  however,  inures  just  as 
surely  to  the  benefit  of  the  shareholders,  and  is  usually 
realizable  through  increased  value  of  the  shares  upon  sale. 
Thus,  dividends  are  not  strictly  revenue,  though  the  share- 
holder may  treat  them  as  such;  his  dividend  may  be  so 
regular  as  practically  to  be  fixed,  or  his  shares  may  be 
preferential,  so  that  to  some  extent  he  is  receiving  an 
ascertained  amount;  or,  as  in  case  of  a  leased  railway,  the 
dividend  may  be  expressly  stipulated  in  a  contract.  Still, 
legally  speaking,  the  dividend  is  instantaneous,  and  does 
not  accrue,  like  interest  and  rent. 

§  8.    Laws  of  Interest 

As  all  investments  are  really  purchases  of  revenue,  and 
as  the  value  of  an  investment  depends  largely  upon  the 
amount  of  revenue  derivable  therefrom,  and  as  the  typical 
form  of  revenue  is  interest,  it  is  necessary  to  study  the  laws 
of  interest,  including  those  more  complex  forms — annuities, 
sinking  funds,  and  amortization.  Although  there  is  a 
special  branch  of  accountancy — ^the  actuarial — which  deals 
not  only  with  these  subjects,  but  with  life  and  other  con- 
tingencies, it  is  yet  very  necessary  for  the  general  accountant 
to  understand  at  least  their  fundamental  principles. 


CHAPTER   II 

INTEREST 

§  g.    Interest 

As  ordinarily  defined,  interest  is  "money  paid  for  the 
use  of  money."  A  better  definition  from  a  mathematical 
standpoint  would  be,  "the  increase  of  indebtedness  through 
lapse  of  time."  Since  the  production  of  additional  wealth  is 
dependent  on  the  processes  of  nature,  and  since  these 
processes  require  time,  it  is  equitable  that  compensation  for 
an  increase  in  time  should  be  made  by  an  increase  in  in- 
debtedness. The  "money  paid"  of  the  first  definition  is  a 
payment  on  account  of  the  general  debt  (including  in- 
terest) ;  the  direct  effect  of  interest  is  to  increase  the  debt, 
while  the  direct  effect  of  a  payment  is  to  reduce  it. 

§  10.    Essentials  of  Interest  Contract 

The  contract,  express  or  implied,  regarding  an  interest 
transaction,  must  take  into  consideration  the  following 
items: 

(1)  Principal.  The  number  of  units  of  value  (dollars, 
pounds,  francs,  marks,  etc.)  originally  loaned  or  invested. 

(2)  Rate.  The  part  of  the  unit  of  value  (usually  a 
small  number  of  hundredths)  which  is  added  to  each  such 
unit  by  the  lapse  of  one  unit  of  time. 

(3)  Frequency.  The  length  of  the  unit  of  time,  measured 
in  years,  months,  or  days.  Weeks  are  not  used  as  time 
units,  nor  are  parts  of  a  day. 

(4)  Time.  The  number  of  units  of  time  during  which 
the  indebtedness  is  to  continue. 

23 


24 


THE  MATHEMATICS  OF  INVESTMENT 


§11.     Interest  Rate 

The  rate  is  usually  spoken  of  as  so  much  per  cent  per 
period,  or  term.  Thus,  if  the  contract  provides  for  the 
payment  of  three  cents  each  year  for  the  use  of  each  dollar 
of  principal,  the  rate  may  be  expressed,  .03  per  annum,  3 
per  centum  per  annum,  3  per  cent,  or  simply  3%.  Where 
the  period  is  not  a  year,  but  a  smaller  unit  of  time,  it  is 
nevertheless  customary  to  speak  of  the  annual  rate.  For 
instance,  instead  of  saying,  "3%  per  half-year,''  we  say, 
"6%,  payable  semi-annually."  In  the  same  way,  1%  per 
quarter  would  be  4%,  payable  quarterly.  In  our  discus- 
sions of  interest,  however,  we  shall  treat  of  periods,  and  of 
the  rate  per  period,  in  order  to  avoid  confusion.  The  in- 
terest rate  will  be  designated  by  the  small  letter  i\  as, 
z  =  .06.  At  the  end  of  the  first  period  the  increased  in- 
debtedness, corresponding  to  the  original  unit  of  indebted- 
ness at  the  beginning  of  the  term,  is  1  +  i  (1.06),  a  very- 
important  quantity  in  computation.  The  subject  of  rates 
of  interest  will  be  discussed  in  greater  detail  in  Chapter 
VIII,  "Nominal  and  Effective  Rates."* 

§  12.     Principal 

Since  each  dollar  increases  just  as  much  as  every  other 
dollar,  the  general  practice  is  to  consider  the  principal  as 
one  dollar  and,  when  the  proper  interest  thereon  has  been 
found,  to  multiply  it  by  the  number  of  dollars. 

§  13.     Simple  and  Compound  Interest 

Interest  is  assumed  to  be  paid  when  due.  If  it  is  not 
so  paid,  it  ought  to  be  added  to  the  principal,  and  interest 
should  be  computed  on  the  increased  principal.  But  the 
law  does  not  directly  sanction  this  compounding  of  interest, 

*For  discussion  of  the  causes  of  higher  or  lower  interest  rates,  see  "The  Rate 
of  Interest,"  by  Prof.  Irving  Fisher. 


INTEREST 


25 


and  simple  interest  is  spoken  of  as  if  it  were  a  distinct 
species  in  which  the  original  principal  remains  unchanged, 
even  though  interest  is  in  default.  There  is  really  no  such 
thing  as  simple  interest,  since  the  interest  money  which  is 
wrongfully  withheld  by  the  borrower,  may  be  by  him  em- 
ployed, and  thus  compound  interest  be  earned.  But  the 
wrong  party  gets  the  benefit  of  the  compounding.  All  the 
calculations  of  finance  depend  upon  compounding  interest, 
which  is  the  only  rational  and  consistent  method.  When 
there  is  occasion  hereafter  to  speak  of  the  interest  for  one 
period,  it  will  be  called  "single  interest." 

§  14.    Punctual  Interest 

The  usual  interest  contract  provides  that  the  increase 
shall  be  paid  off  in  cash  at  the  end  of  each  period,  restoring 
the  principal  to  its  original  amount.  Let  c  denote  the  cash 
payment ;  then  l  +  i  —  c  =  l;  and  the  second  term  would 
repeat  the  same  process.  The  payment  of  cash  for  interest 
must  not  be  regarded  as  the  interest ;  it  is  a  cancellation  of 
part  of  the  increased  principal.  Many  persons,  and  even 
courts,  have  been  misled  by  the  old  definition  of  interest — 
"money  paid  for  the  use  of  money" — into  treating  uncol- 
lected or  unmatured  interest  as  a  nullity,  though  secured  in 
precisely  the  same  way  as  the  principal. 

§  15.     Computation  of  Interest 

But  the  interest  money  may  not  be  paid  exactly  at  the 
end  of  each  term,  either  in  violation  of  the  contract  or  by 
a  special  clause  permitting  it  to  run  on,  or  by  the  debt  being 
assigned  to  a  third  party  at  a  price  which  modifies  the  true 
interest  rate.  In  this  case  the  question  arises :  How  shall 
the  interest  be  computed  for  the  following  periods?  This 
gives  rise  to  a  distinction  between  simple  and  compound 
interest. 


26  THE  MATHEMATICS  OF  INVESTMENT 

§  1 6.     Simple  Interest 

During  the  second  period,  although  the  borrower  has  in 
his  hands  an  increased  principal,  1  +  i,  he  is  at  simple 
interest  charged  with  interest  only  on  1,  and  has  the  free 
use  of  i,  which,  though  small,  has  an  earning  power  pro- 
portionate to  that  of  1.  His  indebtedness  at  the  end  of  the 
second  term  is  1  +  2i,  and  thereafter  1  +  3f,  1  +  4z',  etc. 
After  the  first  period  he  is  not  charged  with  the  agreed  per- 
centage of  the  sum  actually  employed  by  him,  and  this  to 
the  detriment  of  the  creditor.  For  any  scientific  calculation, 
simple  interest  is  impossible  of  application. 

§17.     Compound  Interest 

The  indebtedness  at  the  end  of  the  first  period  is  1  + 1, 
and  up  to  this  point  punctual,  simple,  and  compound  interest 
coincide.  But  in  compound  interest  the  fact  is  recognized 
that  the  increased  principal,  1  +  i,  is  all  subject  to  interest 
during  the  next  period,  and  that  the  debt  increases  by  geo- 
metrical progression,  not  arithmetical.  The  increase  from 
1  to  1  +  i  is  regarded,  not  as  an  addition  of  i  to  1,  but  as  a 
multiplication  of  1  by  the  ratio  of  increase  (1  +  i).  We 
shall  designate  the  ratio  of  increase  by  r  when  convenient, 
although  this  is  merely  an  abbreviation  of  1  +  i,  and  the 
two  expressions  are  at  all  times  interchangeable. 

§  18.     Computation  of  Compound  Interest 

At  the  end  of  the  first  period  (which  is  equivalent  to  the 
beginning  of  the  second  period),  the  actual  indebtedness  is 
1  +  i.  This  amount  is  the  equitable  principal  for  the  second 
period,  and  it  should  be  again  increased  in  the  ratio  1  +  i. 
The  total  indebtedness  at  the  end  of  the  second  period 
(which  is  equivalent  to  the  beginning  of  the  third  period) 
is  therefore  1  X  (1  + 1)  X  (1  +  ?-)•  For  the  sake  of  brevity, 
this  may  be  written  1  X  (1  +  O^j  the  figure  2  (called  an  ex- 


INTEREST 


27 


ponent)  indicating  that  the  expression  (I'+t)  is  to  ,be 
taken  twice  as  a  factor.  Since  the  expression  (1  +  i)  equals 
the  rate,  a  still  simpler  way  of  indicating  the  indebtedness  at 
the  end  of  the  first  period  is  r;  at  the  end  of  the  second 
period,  r^.  At  the  end  of  the  third  period  the  indebtedness 
will  have  become  t^ ;  and  at  the  end  of  period  t,  it  will  have 
become  r\ 

§  19.     Comparison  of  Simple  and  Compound  Interest 

The  following  schedule  shows  the  accumulations  of  in- 
terest for  several  periods,  giving  a  comparison  between  the 
simple  interest  computations  and  the  compound  interest 
computations : 


■ 

Indebtedness 

Indebtedness 

Time 

Based  on 

Based  on 

Simple  Interest 

Compound  Interest* 

Beginning  of  1st  period.. 

1 

1 

Beginning  of  2nd  period. 

1-bi 

l  +  t 

Beginning  of  3rd  period. 

l  +  2i 

(i  +  »T 

Beginning  of  4th  period. . 

l  +  3i 

(i+»r 

Beginning  of  5th  period. . 

l  +  4i 

(1  +  0* 

etc. 

•For  the  benefit  of  students  familiar  with  algebra,  it  may  be  pointed  out  that 
(1  +  O*  =  1  +  2i  + t*.  This  differs  from  the  simple  interest  computation  by  the 
small  quantity  t*.  Similarly,  (1  +  t)^  =  1  +  3t  +  3i^  +  »^,  which  differs  from  the 
simple  interest  result,  1  +  Si,  by  the  quantity  Si'^  +  *"'•  Tests  may  be  readily  made 
of  the  computations  by  substituting  a  numerical  rate,  say  .06,  in  place  of  i.  If  this 
be  done,  the  simple  interest  result  at  the  beginning  of  the  4th  period  is  found  to  be 
14- (3  times  .06),  or  1.18.     The  compound  interest  result  would  be: 

1  =1. 

plus  3  times  .06         =    .18 
plus  3  timco  .06',  or 

3  times  .0036     =    .0108 
plus  .06»  =    .000216 


That  is,  (1.06)«     =  1.191016 


28  THE  MATHEMATICS  OF  INVESTMENT 

§  20.     The  Day  as  a  Time  Unit 

Coming  now  to  a  discussion  of  frequency  and  time,  in 
connection  with  the  subject  of  interest,  we  find  that  the 
smallest  unit  of  time  is  one  day,  since  the  law  does  not 
recognize  interest  for  fractions  of  a  day.  The  legal  day 
begins  at  midnight  and  ends  on  the  following  midnight.  In 
reckoning  from  one  day  to  another,  the  day  from  which 
should  be  excluded.  Thus,  if  a  loan  is  made  at  any  hour 
on  the  third  day  of  the  month  and  is  paid  at  any  hour  on 
the  fourth,  there  is  one  day's  interest  due,  the  interest  being 
for  the  fourth  day  and  not  for  the  third.  Practically  it  is 
the  nights  that  count.  If  five  midnights  have  passed  since 
the  loan  was  made,  then  the  accrued  interest  is  for  a  period 
of  five  days. 

§  21.     The  Month  as  a  Time  Unit 

As  has  been  previously  stated,  weeks  are  not  used  as 
time  units.  The  next  longer  interest  period  after  a  day  is 
a  month.  Calendar  months  are  computed  as  follows :  Com- 
mence at  the  day  from  which  the  reckoning  is  made,  and 
exclude  that  day;  then  the  day  in  the  next  month  having 
the  same  number  will  at  its  close  complete  the  first  month ; 
the  second  month  will  end  with  the  same  numbered  day, 
and  so  on  to  the  same  day  of  the  final  month.  A  difficulty 
arises  in  the  case  where  the  initial  date  is  the  31st,  while  the 
last  month  has  only  thirty  days  or  less.  In  this  case  the 
interest  month  ends  with  the  last  day  of  the  calendar  month. 
For  example,  one  month  from  January  31st,  1912,  was 
February  29th;  one  month  from  January  30th  or  29th,  in 
the  same  year,  also  terminated  on  February  29th ;  in  a  com- 
mon year,  not  a  leap  year,  the  last  day  of  a  period  one 
month  from  January  28th,  29th,  30th  or  31st,  would  be 
February  28th. 


INTEREST 


29 


§  22.     Half  and  Quarter  Years 

Since  there  are  no  fractions  of  a  day  in  interest  compu- 
tations, it  becomes  necessary  to  inquire  what  is  meant  by 
a  half-year  or  by  a  quarter.  In  the  State  of  New  York  the 
Statutory  Construction  Law  (Laws  of  1892,  Chapter  677, 
§  25)  solves  this  difficulty  by  prescribing  that  a  half-year 
is  not  182%  days,  but  six  calendar  months;  and  that  a 
quarter  is  not  91^/4  days,  but  three  calendar  months. 

§  23.    Partial  Interest  Periods 

In  practice  any  fraction  of  an  interest  period  is  com- 
puted at  the  corresponding  fraction  of  the  rate,  although 
theoretically  this  is  not  quite  just.  For  example,  if  the 
interest  rate  is  6%  per  annum,  payable  annually,  making 
the  ratio  of  increase  1.06,  then  it  is  customary  to  consider 
the  ratio  of  increase  for  a  half-year  as  1.03;  whereas 
theoretically  it  should  be  the  square  root  of  1.06,  or  slightly 
over  1.029563. 

If  the  regular  period  is  one  year,  any  odd  days  should 
be  reckoned  as  365ths  of  a  year.  Also,  if  the  contract  is  for 
days  only  and  there  is  no  mention  of  months,  quarters,  or 
half-years,  then  also  a  day  should  be  regarded  as  1/365  of 
a  year.  But  when  the  contract  is  for  months,  quarters,  or 
half-years,  any  fractional  time  should  be  divided  into  months, 
and  there  is  usually  an  odd  number  of  days  left  over.  In 
New  York,  doubt  exists  as  to  how  these  odd  days  should 
be  treated,  whether  on  a  365-day  basis  or  on  a  360-day  basis. 

Before  1892  there  was  no  doubt.  The  statute  distinctly 
stated  that  a  number  of  days  less  than  a  month  should  be 
estimated  for  the  purpose  of  interest  computations  as  30ths 
of  a  month,  or,  consequently,  360ths  of  a  year.  This  was 
a  most  excellent  provision,  and  merely  enacted  what  had 
been  the  custom  long  before.  The  so-called  "360-day"  in- 
terest tables  are  based  upon  this  rule.     In  1892,  however, 


30 


THE  MATHEMATICS  OF  INVESTMENT 


the  revisers  of  the  statutes  of  the  State  of  New  York 
dropped  this  sensible  provision  and  left  the  question  open. 
No  judicial  decision  has  since  been  rendered  on  the  subject, 
but  many  good  lawyers  think  that  the  odd  days  should  be 
computed  as  365ths  of  a  year.  In  business  nearly  every 
one  calls  the  odd  days  360ths,  and  it  is  only  in  legal  account- 
ings that  there  can  be  any  question.  It  would  be  well  if 
the  old  provision  could  be  re-enacted  by  law  or  re-established 
by  the  courts. 

§  24.     Changing  the  Day  Basis 

If  the  interest  for  a  certain  number  of  odd  days  has  been 
computed  on  a  360-day  basis,  a  change  may  be  readily  made 
to  a  365-day  basis  by  subtracting  from  such  interest  1/73 
of  itself.  On  the  other  hand,  if  the  interest  for  an  odd 
number  of  days  has  been  ascertained  on  a  365-day  basis, 
the  addition  of  1/72  of  itself  to  this  amount  will  give  the 
interest  on  a  360-day  basis. 

§  25.     The  Amount — First  Period 

The  principal  and  interest  taken  together  constitute  the 
amount.  At  the  end  of  the  first  half-year  period,  the 
amount  of  $1.00  at  6%  interest,  payable  semi-annually,  is 
$1.03.  Instead  of  considering  the  $1.00  and  the  3  cents  as 
two  separate  items  to  be  added  together,  it  is  best  to  con- 
sider the  operation  as  the  single  one  of  multiplying  $1.00 
by  the  ratio  of  increase,  1.03.  Sometimes  the  error  is  made 
of  considering  that  the  original  principal  of  $1.00  is  multi- 
plied by  $1.03,  or,  in  other  words,  that  a  certain  number 
of  dollars  is  multiplied  by  another  number  of  dollars.  It 
is  well  to  emphasize,  in  this  connection,  the  old  principle 
given  in  arithmetic,  that  one  concrete  number  cannot  be 
multiplied  by  another  concrete  number.  We  cannot  multi- 
ply dollars  by  dollars,  or  feet  by  feet,  or  horses  by  dollars. 
The  multiplicand  may  be  either  a  concrete  or  an  abstract 
number,  but  the  multiplier  must  always  be  abstract. 


INTEREST 


31 


§  26.    The  Amount — Subsequent  Periods 

The  principal  which  is  employed  during  the  second 
period  is  $1.03.  It  is  evident  that  this,  like  the  original 
$1.00,  should  be  multiplied  by  the  ratio  1.03.  The  new 
amount  will  be  the  square  of  1.03,  which  we  may  write: 

1.03  X  1.03 
or,    1.03=^ 
or,    1.0609 

This  is  the  new  amount  on  interest  during  the  third  period. 
At  the  end  of  the  third  period  the  amount  will  be : 

1.03  X  1.03  X  1.03 
or,    1.03' 
or,    1.092727 

At  the  end  of  the  fourth  period  the  amount  becomes : 

1.03* 

or,     1.12550881 

Possibly  at  this  point  the  number  of  decimal  places  may  be 
unwieldy.  If  we  desire  to  have  only  seven  decimal  places, 
we  reject  the  final  1,  rounding  the  result  oif  to  1.1255088; 
if  we  prefer  to  use  only  six  places,  we  round  the  result  up 
to  1.125509,  which  is  more  nearly  correct  than  1.125508. 

§  27.    Exponents  and  Powers 

In  some  of  the  following  paragraphs,  it  will  be  neces- 
sary to  speak  occasionally  of  exponents  and  powers.  In 
the  expression  1.03^,  the  figure  2  is  called  an  exponent,  and 
it  means  (as  indicated  in  the  preceding  paragraph)  that 
1.03  is  to  be  taken  twice  as  a  factor.  In  other  words,  the 
number  is  to  be  multiplied  by  itself.  The  result,  1.0609 
(which  equals  1.03^),  is  said  to  be  the  second  power  of  1.03 ; 
1.092727  is  the  third  power  of  1.03,  and  so  on.  (See  also 
§38.) 


32  THE  MATHEMATICS  OF  INVESTMENT 

§  28.     Finding  the  Amount — Compound  Interest 

The  amount  of  $1.00  at  the  end  of  any  number  of 
periods  is  obtained  by  taking  such  a  power  of  the  ratio  of 
increase  as  is  indicated  by  the  number  of  periods;  or,  in 
other  words,  by  muhiplying  $1.00  by  the  ratio  as  many 
times  as  there  are  periods.  If  the  original  principal  be  sub- 
tracted from  the  amount,  the  remainder  is  the  compound 
interest.  For  example,  in  §  26,  the  amount  of  $1.00  at  the 
ratio  1.03,  for  four  periods,  is  $1.12550881;  and  the  com- 
pound interest  is  $.12550881. 

§  29.    Present  Worth 

The  present  worth  of  a  future  sum  is  a  smaller  sum 
which,  put  at  interest,  will  amount  to  the  future  sum.  The 
present  worth  of  $1.00  is  such  a  sum  as,  at  the  given  rate 
and  for  the  given  period,  will  amount  to  $1.00.  In  order 
to  illustrate  the  method  of  ascertaining  the  present  worth, 
let  us  suppose  that  it  is  desired  to  find  the  present  worth 
of  $1.00,  due  in  four  years,  the  ratio  of  increase  being  1.03 
per  annum.  The  required  figure  must  evidently  be  such 
that,  when  multiplied  four  times  in  succession  by  1.03,  the 
result  will  be  $1.00.  Therefore,  by  using  the  reverse 
process,  division,  the  required  figure  may  be  obtained.  The 
first  operation,  by  ordinary  long  division,  results  as  follows : 

1.03  )   1.00000000   (  .970873 
927 
730 
721 
900 
824 
760 
721 
390 
309 
81 


INTEREST  33 

The  result,  rounded  up  at  the  6th  place,  is  .970874,  this 
being  the  present  worth  of  $1.00  due  in  one  period  at  3% 
interest.  The  present  worth  for  two  periods  may  be  ob- 
tained either  by  again  dividing  .970874  by  1.03,  or  by  mul- 
tiplying .970874  by  itself,  or  by  dividing  1  by  1.0609  (the 
square  of  1.03),  each  of  which  operations  gives  the  same 
result,  .942596.  The  present  worth  for  three  periods  may 
also  be  obtained  in  several  ways,  the  result  being  the  same 
in  all  cases,  .915142,  or  ^^.  The  present  worth  for  four 
periods  is  ^,  or  .888487. 

§  30.     Present  Worth  and  Amount  Series 

If  we  arrange  these  results  in  reverse  order,  followed 
by  $1.00  and  by  the  amounts  computed  in  §  26,  we  have  a 
continuous  series : 


1- 

- 1.03* 

-=  .888487 

1- 

- 1.03' 

-=  .915142 

1- 

-1.03' 

-=  .942596 

1- 

-1.03 

=-  .970874 

1. 
=  1.03 

1  X  1.03 

1  X  1.03' 

=  1.0609 

1  X  1.03' 

=  1.092727 

1> 

<  1.03* 

=  1.125509 

§  31.     Relation  between  Present  Worth  and  Amount 

In  the  foregoing  series,  which  might  be  extended  in- 
definitely upward  and  downward,  every  term  is  a  present 
worth  of  the  one  which  immediately  follows  it,  and  an 
amount  of  the  one  which  immediately  precedes  it.  When 
one  number  is  the  amount  of  another,  the  latter  number  is 
the  present  worth  of  the  former.  For  example,  .888487  is 
the  present  worth  of  1.125509  for  8  interest  periods;  and, 
on  the  other  hand,  1.125509  is  the  amount  of  .888487,  for 
the  same  number  of  interest  periods  and  at  the  same  ratio. 


34 


THE  MATHEMATICS  OF  INVESTMENT 


In  some  instances  in  this  series,  a  present  worth  and  its 
corresponding  amount  are  reciprocals  (that  is,  their  product 
is  1),  but  this  is  true  only  when  the  two  figures  are  distant 
an  equal  number  of  periods  from  1,  the  present-worth 
figure  being  upward  from  1  and  the  amount  figure  being 
downward  from  1.  Thus,  .915142,  or  1  -^- 1.03^,  is  the 
reciprocal  of  1.092727,  or  1.03^ 

§  32.     Formation  of  Series 

K  any  term  of  the  series  be  multiplied  by  1.03,  the 
product  will  be  the  next  following  term ;  if  it  be  divided  by 
1.03  or  (which  amounts  to  the  same  thing)  be  multiplied  by 
.970874,  the  result  will  be  the  next  preceding  term.  Since 
multiplying  by  1.03  is  easier  than  dividing  by  it,  and  also 
easier  than  multiplying  by  .970874,  the  easiest  way  of 
obtaining  the  different  numbers  in  the  series  is  to  compute 
first  the  smallest  number  (in  this  case,  .888487),  and  then 
perform  successive  multiplications  by  1.03.  A  brief  process 
for  finding  this  initial  number  will  be  explained  in  the  next 
chapter. 

§  33.     Discount 

In  considering  the  present  worth  of  $1.00  for  a  single 
period  (.970874),  it  is  evident  that  the  original  $1.00  has 
been  diminished  by  .029126,  which  is  a  little  less  than  .03; 
in  fact  it  is  .03  -^  1.03.  This  difference,  .029126,  is  called 
the  discount.  In  the  present  worth  for  two  periods,  the  dis- 
count is  1  — .942596,  or  .057404.  This  discount  for  two 
periods,  and  likewise  the  discounts  for  three  or  more 
periods,  are  called  compound  discounts. 

§  34.     Computing  Compound  Discount 

The  compound  discount  for  any  number  of  periods  may 
be  found  either  by  subtracting  the  present  worth  from  1,  or 


INTEREST 


35 


by  finding  the  present  worth  of  the  compound  interest  for 
the  same  time  and  at  the  same  rate.  As  an  illustration, 
suppose  that  it  is  desired  to  find  the  compound  discount  of 
$1.00  for  three  periods  at  3%.  First,  we  may  subtract  the 
present  worth,  .915142,  from.  1,  which  gives  the  compound 
discount  as  .084858.  Second,  we  may  divide  the  compound 
interest  (.092727)  by  the  amount  of  $1.00  for  the  three 
periods  (1.092727),  which  gives  the  compound  discount  the 
same  as  before,  .084858. 

§  35.     Formulas  for  Interest  Calculations 

We  may  reduce  the  rules  to  more  compact  form  by  the 
use  of  symbols.  Let  a  represent  the  amount  of  $1.00  for 
any  number  of  periods  (it  periods) ;  p  the  present  worth; 
i  the  rate  of  interest  per  period ;  d  the  rate  of  discount  per 
period,  and  n  the  number  of  periods.  Let  the  compound 
interest  be  represented  by  I,  and  the  compound  discount  by 
D.  Then,  by  §17,  the  ratio  of  increase  is  (1  +  i).  By 
§28,  a=(l-}-iy;  and  I==a  — 1.  By  §29,  />  =  1-^ 
(1  +  0";  and  by  §  34,  D  =  1  -  /),  or  I  -^  a. 

§  36.     Use  of  Logarithms 

The  method  of  ascertaining  the  values  of  a  and  p 
through  successive  multiplications  and  divisions  for  a  large 
number  of  periods,  is  intolerably  slow.  A  much  briefer  way 
is  by  the  use  of  certain  auxiliary  numbers  called  logarithms 
as  explained  in  the  next  chapter. 


CHAPTER  III 

THE   USE   OF   LOGARITHMS 

§  37.    Purpose  of  Logarithms 

For  multiplying  or  dividing  a  great  many  times  by  the 
same  number,  or  for  finding  powers  and  roots,  there  is  no 
device  superior  to  a  table  of  logarithms.  Although  the 
computation  of  logarithms — as  in  the  formation  of  a  table  of 
logarithms — requires  a  knowledge  of  algebra,  the  practical 
use  of  logarithmic  tables  does  not  require  such  knowledge. 
The  aid  derived  from  such  tables  is  purely  arithmetical,  and 
the  occasional  prejudice  against  logarithms  as  something 
mysterious  or  occult  is  without  reasonable  foundation. 

§  38.    Exponents,  Powers,  and  Roots 

We  have  seen  in  §  27  that  an  exponent  is  a  number  writ- 
ten at  the  right  and  slightly  above  another  number  to  indi- 
cate how  many  times  the  latter  is  to  be  taken  as  a  factor; 
and  also  that  a  power  is  the  result  obtained  by  taking  any 
given  number  a  certain  number  of  times  as  a  factor.  We 
now  add  that  a  root  is  the  number  repeated  as  a  factor  to 
form  a  power.  The  following  table  exemplifies  roots, 
exponents,  and  powers : 

Roots  and  ^^ 

T,  ,  Powers 

Exponents 

22  =  2X2         =4 

32  =  3x3         =9 

4'  =  4  X  4         =16 

etc. 
2^  =  2X2X2=  8 
3' =  3X3X3  =  27 
43  =  4X4X4  =  64 

etc. 

36 


THE  USE  OF  LOGARITHMS 


37 


The  root  of  a  number  is  called  its  first  power.  When  the 
root  is  taken  twice  as  a  factor,  the  result  is  called  the  second 
power,  or  square ;  when  taken  three  times,  the  result  is  the 
third  power,  or  cube;  we  may  in  like  manner  obtain  the 
fourth,  fifth,  or  any  power  of  a  root  by  repeating  it  as  a 
factor  the  required  number  of  times. 

§  39.    Logarithms  as  Exponents 

Now,  logarithms  are  merely  exponents  of  certain  roots 
which  are  called  bases.  The  common  system  of  logarithms 
is  based  upon  the  number  10,  this  number  being  the  basis  of 
our  decimal  system  of  numeration. 

Taking  a  specific  illustration,  let  us  multiply  six  lO's 
together,  10  X  10  X  10  X  10  X  10  X  10 ;  we  may  write  the 
result  as : 

1,000,000 
or,  10" 
or,  the  sixth  power  of  ten. 

The  small  figure  "6'*  is  the  exponent  of  the  power.  A 
series  of  some  of  the  powers  of  10  might  be  represented  as 
follows : 

1,000,000  or  10" 


100,000 

10° 

10,000 

10* 

1,000 

10' 

100 

10^ 

10 

10' 

1 

tt 

10" 

From  the  above  series,  the  following  observations  may 
be  made : 

(1)  The  number  of  zeroes  in  any  number  in  the  first 
column  is  the  same  as  the  exponent  in  the  second  column. 

(2)  Each  term  in  the  first  column  is  one-tenth  of  the 


2fi  THE  MATHEMATICS  OF  INVESTMENT 

one  above  it,  while  in  the  second  column  each  exponent  is  one 
less  than  the  exponent  above  it.  This  leads  to  the  result 
that  10*^  =  1,  which  at  first  seems  impossible.  It  is  difficult 
to  understand  how  10,  taken  zero  times  as  a  factor,  equals 
1,  but  such  nevertheless  is  a  fact,  as  can  be  easily  demon- 
strated by  algebra.* 

(3)  By  adding  any  two  exponents  in  the  second  column, 
w^e  may  find  the  result  of  multiplying  together  the  two 
corresponding  numbers  in  the  first  column.  For  example, 
10'  (or  100)  times  10'  (or  1,000)  equals  10' +  ^  i.e.,  10' 
(or  100,000)  ;  in  other  words,  by  adding  the  logarithms  of 
two  numbers  we  obtain  the  logarithm  of  their  product. 
Again,  by  finding  the  difference  between  any  two  logarithms 
in  the  second  column,  we  may  find  the  quotients  of  the 
corresponding  numbers  in  the  first  column.  For  example, 
10'  - '  =  10'  =  1,000,  which  =  100,000  -f- 100;  i.e.,  by  sub- 
tracting logarithms  the  logarithms  of  quotients  are  found. 

Suppose  we  should  wish  to  obtain  the  second  power  of 
10';  the  exponent  (or  index)  of  the  second  power  is  2;  and 
10^  X  2  ==  ;^Q6  ^  -^QQQ  ^  -^QQQ^  ^j.  ^^000,000 ;  from  which  it 

appears  that  by  multiplying  the  logarithm  3  by  the  index  2 
we  have  obtained  the  square  of  10',  or  1,000.  Again  10^  "^  ^ 
=  10' =  1,000,  or  the  square  root  of  1,000,000;  from 
which  it  appears  that,  by  dividing  the  logarithm  6  by  the 
index  2,  we  obtain  the  square  root  of  10^. 

§  40.     Rules  and  Symbols  of  Logarithms 

Summarized  very  briefly,  the  rules  of  logarithms,  de- 
duced from  the  foregoing  illustrations,  are  as  follows: 


By  the  use  of  the  equation : 

Therefore,  x<>  =    1 

or,  if  X  =10 

10»=    1 


—  =  !•   —  =x*-»  — x» 


THE  USE  OF  LOGARITHMS  39 

(1)  By  adding  logarithms,  numbers  are  multiplied. 

(2)  By  subtracting  logarithms,  numbers  are  divided. 

(3)  By  midtiplying  logarithms,  numbers  are  raised  to 
powers. 

(4)  By  dividing  logarithms,  the  roots  of  numbers  are 
extracted. 

The  last  two  of  these  rules  are  the  only  ones  necessary 
to  be  employed  in  the  calculations  of  compound  interest. 

With  this  preliminary  explanation  of  logarithms,  the 
series  in  §  39  may  be  rewritten  and  "extended";  the  symbol 
nl  meaning  *'the  number  whose  logarithm  is." 

The  base  being  10, 
1,000,000  is  the  number  whose  logarithm  is  6, 
or,  in  contracted  form. 


10«  = 

1,000,000. 

nl 

6 

10'  = 

100,000. 

nl 

5 

10*  = 

10,000. 

nl 

4 

10'  = 

1,000. 

nl 

3 

10"  = 

100. 

nl 

2 

10^  = 

10. 

nl 

1 

10«  = 

1. 

nl 

0 

.1 

nl  ■ 

-1 

.01 

nl 

-2 

.001 

nl 

-3 

.0001 

nl 

-4 

.00001 

nl  ■ 

-5 

Occasionally  the  symbol  In  will  also  be  used,  its  meaning 
being  *'the  logarithm  of  the  number." 

From  the  above,  it  will  be  seen  that  the  logarithm  of  a 
number  is  merely  the  exponent  which  indicates  the  power 
to  which  some  given  number,  called  the  base,  would  have 
to  be  raised  in  order  to  give  that  number.     If  the  number 


40 


THE  MATHEMATICS  OF  INVESTMENT 


were  100  and  the  base  10,  then  10  would  have  to  be  raised 
to  the  second  power  to  give  100;  in  other  words,  the  loga- 
rithrn  of  100,  with  10  as  a  base,  is  2.  If  the  number  were 
100,000,  the  base  still  being  10,  then  10  would  have  to  be 
raised  to  the  fifth  power  to  give  the  number;  or  we  may 
say,  in  different  language,  that  the  logarithm  of  100,000, 
with  10  as  a  base,  is  5. 


§  41.     The  Two  Parts  of  a  Logarithm 

The  logarithms  of  a  few  numbers  have  already  been 
given,  but  for  practical  use  in  calculating  we  need  the  loga- 
rithms of  a  great  many  others.  From  the  series  in  §  39,  it 
may  be  readily  inferred  that  the  numbers  between  1  and  10 
must  have  their  logarithms  between  0  and  1 ;  that  is,  the 
logarithms  of  these  numbers  must  be  fractions.  In  tables 
of  logarithms,  these  fractions  are  expressed  as  decimals, 
the  usual  number  of  decimal  places  being  seven.  Similarly, 
the  numbers  between  10  and  100  have  their  logarithms  be- 
tween 1  and  2 ;  that  is,  these  logarithms  are  1  plus  a  decimal 
fraction. 

To  give  a  few  illustrations: 

The  logarithm  of     10  =  1. 

12  =  1.0792 
20  =  1.3010 
50  =  1.6990 
90  =  1.9542 
99  =  1.9956 
100  =  2. 

It  will  be  observed  that  there  are  usually  two  parts  to  a 
logarithm — ^the  decimal  part  and  the  whole  number  preced- 
ing the  decimal.  The  decimal  part  is  known  as  the  mantissa, 
and  the  whole  number  as  the  characteristic.  From  the  man- 
tissa of  a  logarithm,  we  are  able  (through  the  aid  of  loga- 


THE  USE  OF  LOGARITHMS 


41 


rithmic  tables)  to  determine  the  corresponding  number, 
except  as  to  the  position  of  its  decimal  point.  This  latter 
is  determined  by  the  characteristic,  which,  for  numbers 
greater  than  1,  is  always  one  less  than  the  number  of  places 
to  the  left  of  the  decimal  point.  For  numbers  less  than  1, 
the  characteristic  is  said  to  be  negative,  and  it  is  equal  to 
the  number  of  places  to  the  right  from  the  decimal  point  to 
the  place  occupied  by  the  first  significant  figure  of  the 
decimal.  A  negative  characteristic  is  indicated  by  a  short 
dash  placed  above  it.  A  characteristic  may  thus  be  either 
positive  or  negative,  but  a  mantissa  is  always  positive. 

§42.     Mantissa  Not  Affected  by  Position  of  the  Decimal 
Point 

In  the  logarithms  of  20,  200,  2,000,  20,000,  200,000, 
2,000,000,  etc.,  we  shall  find  the  same  decimal  part,  .301  030 
(which  is  the  logarithm  of  2),  preceded  by  the  figures  1,  2, 
3,  4,  5,  6,  etc.  This  same  thing  is  true  of  any  combination 
of  figures;  that  is  to  say,  whatever  may  be  the  position  of 
the  decimal  point  in  a  number,  the  logarithm  of  that  num- 
ber always  has  the  same  decimal  fraction,  or  mantissa. 

Thus,  if  the  logarithm  of  2.378  is  .376  212,  then, 

.0002378  nl  4.376  212 
.002378     nl  3.376  212 


.02378 

nl   2.376  212 

.2378 

nl   1.376  212 

2.378 

nl    .376  212 

.23.78 

nl   1.376  212 

237.8 

nl   2.376  212 

2,378. 

nl   3.376  212 

23,780. 

nl  4.376  212 

237,800. 

nl   5.376  212 

etc. 

etc. 

42 


THE  MATHEMATICS  OF  INVESTMENT 


§  43.     Four-Place  Table  of  Logarithms 

Illustrations  will  now  be  given  of  the  properties  of  loga- 
rithms, and  for  this  purpose  a  table  of  the  logarithms  of 
numbers  from  10  to  99,  inclusive,  to  four  places  of  decimals, 
is  given  on  the  following  pages.  This  is  a  very  simple  table 
of  logarithms,  and  is  known  as  a  four-place  table. 

-The  ordinary  tables  of  logarithms  are  calculated  to  seven 
places  of  decimals.  If  it  is  desired  to  multiply  the  number 
82  by  1.03  fifty  times  in  succession,  ordinary  logarithm 
tables  would  give  only  the  first  seven  figures  of  the  answer. 
If  this  operation  were  performed  accurately  by  the  simple 
processes  of  multiplication,  the  answer  vrould  contain  103 
figures,  3  in  front  of  the  decimal  point  and  100  after  it. 
Since  the  figures  after  the  first  seven  are  for  most  purposes 
insignificant,  the  result  obtained  by  logarithms  will  be  near 
enough  even  if  rounded  off  at  the  sixth  figure. 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

0000 

043 

086 

128 

170 

212 

253 

294 

334 

374 

11 

0414 

453 

492 

531 

569 

607 

645 

682 

719 

755 

12 

0792 

828 

864 

899 

934 

969 

*004 

*038 

*072 

♦106 

13 

1139 

173 

206 

239 

271 

303 

335 

367 

399 

430 

14 

1461 

492 

523 

553 

584 

614 

644 

67Z 

703 

732 

15 

1761 

790 

818 

847 

875 

903 

931 

959 

987 

*014 

16 

2041 

068 

095 

122 

148 

175 

201 

227 

253 

279 

17 

2304 

330 

355 

380 

405 

430 

455 

480 

504 

529 

18 

2553 

577 

601 

625 

648 

672 

695 

718 

742 

765 

19 

2788 

810 

833 

856 

878 

900 

923 

945 

967 

989 

20 

3010 

032 

054 

075 

096 

118 

139 

160 

181 

201 

21 

3222 

243 

263 

284 

304 

324 

345 

365 

385 

404 

22 

3424 

444 

464 

483 

502 

522 

541 

560 

579 

598 

23 

3617 

636 

655 

674 

692 

711 

729 

747 

766 

784 

24 

3802 

820 

838 

856 

874 

892 

909 

927 

945 

962 

25 

3979 

997 

*014 

*031 

*048 

*065 

*082 

*099 

*116 

*133 

26 

4150 

166 

183 

200 

216 

232 

249 

265 

281 

298 

27 

4314 

330 

346 

362 

378 

393 

409 

425 

440 

456 

28 

4472 

487 

502 

518 

533 

548 

564 

579 

594 

609 

29 

4624 

639 

654 

669 

683 

698 

713 

728 

742 

757 

See  explanation  at  the  end  of  this  section. 


THE  USE  OF  LOGARITHMS 


43 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

30 

4771 

786 

800 

814 

829 

843 

857 

871 

886 

900 

31 

4914 

928 

942 

955 

969 

983 

997 

*011 

*024 

*038 

32 

5051 

065 

079 

092 

105 

119 

132 

145 

159 

172 

33 

5185 

198 

211 

224 

237 

250 

263 

276 

289 

302 

34 

5315 

328 

340 

353 

366 

378 

391 

403 

416 

428 

35 

5441 

453 

465 

478 

490 

502 

514 

527 

539 

551 

36 

5563 

575 

587 

599 

611 

623 

635 

647 

658 

670 

37 

5682 

694 

705 

717 

729 

740 

752 

763 

775 

786 

38 

5798 

809 

821 

832 

843 

855 

866 

877 

888 

899 

39 

5911 

922 

933 

944 

955 

966 

977 

988 

999 

*010 

40 

6021 

031 

042 

053 

064 

075 

085 

096 

107 

117 

41 

6128 

138 

149 

160 

170 

180 

191 

201 

212 

222 

42 

6232 

243 

253 

263 

274 

284 

294 

304 

314 

325 

43 

6335 

345 

355 

365 

375 

385 

395 

405 

415 

425 

44 

6435 

444 

454 

464 

474 

484 

493 

503 

513 

522 

45 

6532 

542 

551 

561 

571 

580 

590 

599 

609 

618 

46 

6628 

637 

646 

656 

665 

675 

684 

693 

702 

712 

47 

6721 

730 

739 

749 

758 

767 

776 

785 

794 

803 

48 

6812 

821 

830 

839 

848 

857 

866 

875 

884 

893 

49 

6902 

911 

920 

928 

937 

946 

955 

964 

972 

981 

50 

6990 

998 

^007 

*016 

*024 

*033 

*042 

*050 

*059 

*067 

51 

7076 

084 

093 

101 

110 

118 

126 

135 

143 

152 

52 

7160 

168 

177 

185 

193 

202 

210 

218 

226 

235 

53 

7243 

251 

259 

267 

275 

282 

292 

300 

308 

316 

54 

7324 

332 

340 

348 

356 

364 

372 

380 

388 

396 

55 

7404 

412 

419 

427 

435 

443 

451 

459 

466 

474 

56 

7482 

490 

497 

505 

513 

520 

528 

536 

543 

551 

57 

7559 

566 

574 

582 

589 

597 

604 

612 

619 

627 

58 

7634 

642 

649 

657 

664 

672 

679 

686 

694 

701 

59 

7709 

716 

723 

731 

738 

745 

752 

760 

767 

774 

60 

7782 

789 

796 

803 

810 

818 

825 

832 

839 

846 

61 

7853 

860 

868 

875 

882 

889 

896 

903 

910 

917 

62 

7924 

931 

938 

945 

952 

959 

966 

973 

980 

987 

63 

7993 

*000 

*007 

*014 

*021 

*028 

*035 

*041 

*048 

*055 

64 

8062 

069 

075 

082 

089 

096 

102 

109 

116 

122 

*  See  explanation  at  the  end  of  this  section. 


44 


THE  MATHEMATICS  OF  INVESTMENT 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

65 

8129 

136 

142 

149 

156 

162 

169 

176 

182 

189 

66 

8195 

202 

209 

215 

222 

228 

235 

241 

248 

254 

67 

8261 

267 

274 

280 

287 

293 

299 

306 

312 

319 

68 

8325 

331 

338 

344 

351 

357 

363 

370 

376 

382 

69 

8388 

395 

401 

407 

414 

420 

426 

432 

439 

445 

70 

8451 

457 

463 

470 

476 

482 

488 

494 

500 

506 

71 

8513 

519 

525 

531 

537 

543 

549 

555 

561 

567 

72 

8573 

579 

585 

591 

597 

603 

609 

615 

621 

627 

73 

8633 

639 

645 

651 

657 

663 

669 

675 

681 

686 

74 

8692 

698 

704 

710 

716 

722 

727 

7ZZ 

739 

745 

75 

8751 

756 

762 

768 

774 

779 

785 

791 

797 

802 

76 

8808 

814 

820 

825 

831 

837 

842 

848 

854 

859 

77 

8865 

871 

876 

882 

887 

893 

899 

904 

910 

915 

78 

8921 

927 

932 

938 

943 

949 

954 

960 

965 

971 

79 

8976 

982 

987 

993 

998 

*004 

*009 

*015 

*020 

*025 

80 

9031 

036 

042 

047 

053 

058 

063 

069 

074 

079 

81 

9085 

090 

096 

101 

106 

112 

117 

122 

128 

133 

82 

9138 

143 

149 

154 

159 

165 

170 

175 

180 

186 

83 

9191 

196 

201 

206 

212 

217 

222 

227 

232 

238 

84 

9243 

248 

253 

258 

263 

269 

274 

279 

284 

289 

85 

9294 

299 

304 

309 

315 

320 

325 

330 

335 

340 

86 

9345 

350 

355 

360 

365 

370 

375 

380 

385 

390 

87 

9395 

400 

405 

410 

415 

420 

425 

430 

435 

440 

88 

9445 

450 

455 

460 

465 

469 

474 

479 

484 

489 

89 

9494 

499 

504 

509 

513 

518 

523 

528 

533 

538 

90 

9542 

547 

552 

557 

562 

566 

571 

576 

581 

586 

91 

9590 

595 

600 

605 

609 

614 

619 

624 

628 

633 

92 

9638 

643 

647 

652 

657 

661 

666 

671 

675 

680 

93 

9685 

689 

694 

699 

703 

708 

713 

717 

722 

727 

94 

9731 

736 

741 

745 

750 

754 

.  759 

763 

768 

77Z 

95 

9777 

782 

786 

791 

795 

800 

805 

809 

814 

818 

96 

9823 

827 

832 

836 

841 

845 

850 

854 

859 

863 

97 

9868 

872 

877 

881 

886 

890 

894 

899 

903 

908 

98 

9912 

917 

921 

926 

930 

934 

939 

943 

948 

952 

99 

9956 

961 

965 

969 

974 

978 

983 

987 

991 

996 

See  explanation  at  the  end  of  this  section. 


THE  USE  OF  LOGARITHMS  .45 

In  the  table  preceding,  the  figures  in  the  column  headed 
"N"  denote  the  numbers  whose  logarithms  are  given.  These 
numbers  must  be  considered  in  conjunction  with  the  num- 
bers at  the  top  of  the  remaining  ten  columns ;  in  other  words, 
we  can  find  the  logarithm  not  only  of  the  number  (say)  34, 
but  also  of  34.1,  34.2,  34.3,  etc.,  and  similarly  of  .34,  .341, 
.342,  .343,  etc.,  and  of  3,400,  3,410,  3,420,  3,430,  etc. 

The  figures  in  the  columns  headed  "0,"  "1,''  "2,"  etc., 
represent  simply  the  decimal  parts  (or  mantissas)  of  the 
logarithms;  the  whole  (or  integral)  parts  of  the  logarithms 
must  always  be  determined  by  inspection  (§41). 

The  column  headed  "0"  has  four  places  of  figures,  while 
the  following  columns  have  only  three  places.  This  is  done 
to  save  space,  since  a  fourth  figure  is  assumed  to  be  pre- 
fixed, this  figure  being  the  same  as  the  first  figure  in  the 
four-place  column.  There  is  an  exception  to  this  rule  in  the 
case  of  figures  prefixed  by  an  asterisk,  and  three  or  four 
examples  will  serve  to  make  this  clear: 

Corresponding 
Number  Logarithm 

2.50  .3979 

2.51  .3997 

2.52  .4014  (not   .3014) 

2.53  .4031  (not   .3031) 
etc.  etc. 


§  44.     Multiplication  by  Logarithms 

As  stated  in  §  40,  there  are  four  general  rules  regarding 
logarithms,  and  these  will  now  be  illustrated  in  order. 

Rule  1:  The  sum  of  the  logarithms  of  two  or  more 
numbers  is  the  logarithm  of  their  product. 


46  THE  MATHEMATICS  OF  INVESTMENT 

2  nl       .3010 

3  nl       .4771 


2X3 

6  nl 

.778t 

4  nl 

.6021 

14  nl 

1.1461 

4X14 

56  nl 

1.7482 

5  w/ 

.6990 

20  nl 

1.3010 

6X20 

100  w/ 

2.0000 

In  these  and  other  illustrations,  there  may  be  apparent 
errors  in  the  final  decimal  figure,  due  to  throwing  away  or 
adding  on  parts  of  decimals,  as  the  case  might  be.  For 
example,  in  the  first  illustration  given  above,  the  logarithm 
of  2  to  six  places  is  .301030,  and  the  logarithm  of  3  is 
.477121;  the  logarithm  of  6,  the  product,  is  .778151. 

§  45.     Division  by  Logarithms 

The  logarithm  of  a  product  is  obtained  by  finding  the 
sum  of  the  logarithms  of  the  factors,  and,  as  division  is  the 
converse  of  multiplication,  the  logarithm  of  a  quotient  is 
obtained  by  finding  the  difference  between  the  logarithms 
of  the  dividend  and  divisor. 

Rule  2 :  The  difference  of  the  logarithms  of  two  num- 
bers is  the  logarithm  of  their  quotient. 

Required  the  quotient  of  6  -^  2. 
6     nl      .7782 
2     nl       .3010 
6-^2     nl      .4771     In    3 

Required  the  quotient  of  42  -^  14. 
42     nl     1.6232 
14     nl     1.1461 
42-^14    nl      .4771     In    3 


THE  USE  OF  LOGARITHMS  47 

Required  the  quotient  of  100  -^-  4. 
100     nl     2.0000 
4     nl       .6021 
100-^4     fil     1.3979     In    25 

§  46.    Powers  by  Logarithms 

Rule  3  :  The  logarithm  of  the  power  of  a  number  is  equal 
to  the  logarithm  of  the  number  multiplied  by  the  exponent 
of  the  power. 

Let  it  be  required  to  find  the  third  power  of  2,  that  is, 
2',  or  the  product  of  2  X  2  X  2. 

2      nl     .3010 
2'     w/     3  X  .3010 

or,  .9030     In     8 

Required  the  fourth  power  of  5,  that  is,  5*. 
5      nl     .6990 
5'     nl     4  X. 6990 

or,  2.7960     In     625 

It  has  been  observed  in  this  connection  (§  38),  that  the 
second  power  is  usually  called  the  square,  and  the  third 
power  the  cube. 

§  47.     Roots  by  Logarithms 

The  fourth  general  rule  regarding  logarithms  refers  to 
the  extraction  of  roots.  We  have  seen  in  §38  that,  if  a 
certain  number  is  a  power  of  another,  we  call  the  latter 
number  a  root  of  the  former.  For  example,  since  2  X  2  X 
2  X  2  X  2  =  32,  it  is  said  that  32  is  the  5th  power  of  2,  and 
that  the  5th  root  of  32  is  2.  The  usual  way  of  expressing 
this  latter  fact  is : 

V32  =  2 
or,  32^  =  2 


48  THE  MATHEMATICS  OF  INVESTMENT 

With  the  above  explanation,  the  fourth  rule  is  now 
stated,  and  it  will  be  observed  that  it  is  the  converse  of  the 
third  rule. 

Rule  4 :  The  logarithm  of  the  root  of  a  number  is  equal 
to  the  logarithm  of  the  number  divided  by  the  index  of  the 
root. 

As  an  illustration,  let  it  be  required  to  find  the  square 
root  of  49. 

49   •    nl     1.6902 
V49,     or  49'^     nl     1/2  of  1.6902 

or,  .8451     In     7 

Required  the  cube  root  of  512. 

512        nl     2.Y093 


V512,  or  512^     nl     Ys  of  2.7093 

or,  .9031     In     8 

§  48.    Fractional  Exponents 

Such  an  exponent  as  %  requires  explanation.  It  signi- 
fies the  third  power  of  the  fourth  root,  or  the  fourth  root 
of  the  third  power.  Thus,  the  value  of  10^*  may  be  ascer- 
tained by  finding  the  fourth  root  of  10,  and  then  getting  the 
cube  of  this  root;  or  by  finding  the  cube  of  10,  which  is 
1,000,  and  then  taking  the  fourth  root  (or  the  square  root 
of  the  square  root)  of  1,000.  By  the  methods  of  arithmetic, 
the  value  of  lO''^  is  thus  found  to  be  5.62+;  or,  in  other 
words,  the  logarithm  of  5.62  is  approximately  .75.  It  is 
interesting  to  compare  this  result  with  the  table  in  §  43, 
where  it  is  indicated  that  the  logarithm  of  5.62  is  .7497, 
which  is  very  close  to  .7500.  Fractional  exponents  may  be 
expressed  as  decimal,  instead  of  common,  fractions ;  and,  in 
fact,  that  is  what  most  logarithms  are :  simply  fractional 
exponents  of  10,  expressed  decimally. 


THE  USE  OF  LOGARITHMS 


49 


§  49.     Use  of  Logarithms  in  Computing  Compound  Interest 

To  demonstrate  the  use  of  logarithms  in  compound  in- 
terest, let  us  take  an  example  and  work  it  out,  illustrating 
each  step.  We  will  take  3%  as  the  rate,  the  same  as  already 
used  (§§25-30),  but  endeavor  to  find  the  amount  for  50 
periods,  instead  of  for  4  periods. 

The  ratio  of  increase  is  1.03.  Looking  for  the  logarithm 
(to  eight  decimal  places)  of  this  ratio  (Chambers'  or  Bab- 
bage's  tables,  page  192)  we  find  this  line : 

No.  0123456789 

10300  0128  3722  4144  4566  4987   5409  5831   6252  6674  7096   7517 

The  meaning  of  this  line  is  that  the  logarithms  are  as 
follows : 


1.03 

nl 

.01283722 

1.03001 

nl 

.01284144 

1.03002 

nl 

.01284566 

1.03003 

nl 

.01284987, 

1.03009     nl     .01287517 

The  first  figures  of  both  the  numbers  and  the  logarithms 
are  given  only  once  in  the  table,  which  saves  space  in  print- 
ing and  time  in  searching. 

Since  1.03  is  to  be  taken  as  a  factor  50  times,  we  must 
multiply  its  logarithm  by  50,  as  stated  in  Rule  3  (§46). 
This  gives : 

50  X.01283722  =  .6418610 

The  result  is  the  logarithm  of  the  answer;  for,  when  we 
have  found  the  corresponding  number,  we  shall  know  the 
value  of  1.03^^ 

We  must  now  look  in  the  right-hand  columns  for  the 
logarithm  figures  .6418610.  We  first  look  for  the  641, 
which   stands   out   by   itself,    overhanging   a   blank   space 


so 


THE  MATHEMATICS  OF  INVESTMENT 


(Chambers'  or  Babbage's  tables,  page  73),  and  we  find  that 

the  nearest  approach  to  .6418610  is  .6418606,  which  latter 

is  indicated  as  the  logarithm  of  the  number  4.3839.     The 

next  nearest  logarithm  is  .6418705,  which  corresponds  to 

the  number  4.3840.     The  following  tabulation  shows  the 

details  more  clearly : 

Corresponding 
Logarithm  Number 

.6418606  4.3839 

.6418610       To  be  determined 

.6418705  4.3840 

It  is  evident  that  the  number  to  be  determined  lies  be- 
tween 4.3839  and  4.3840,  which  differ  by  .0001.  The  dif- 
ference between  the  first  and  third  logarithms  is  .0000099, 
and  between  the  first  and  second  logarithms  is  .0000004. 
For  practical  purposes,  we  take  4/99  of  the  difference 
between  the  numbers  (.0001),  and  add  this  amount  to 
the  smaller  number,  thus  obtaining  the  required  number 
4.383904.  In  order  to  assist  in  determining  the  decimal 
value  of  4/99  and  similar  fractions,  little  difference-tables 
are  usually  given  in  the  margins  of  the  pages  of  logarithm 
tables,  the  table  for  99  reading  as  follows: 

99 


1 

10 

2 

20 

3 

30 

4 

40 

5 

50 

6 

59 

7 

69 

8 

79 

9 

89 

The  meaning  of  this  table  is  that  40/99  =  .40;  4/99  = 


THE  USE  OF  LOGARITHMS  5I 

.04;  8/99  =  .079 ;  etc.    By  the  use  of  these  small  tables,  the 
labor  of  dividing  is  thus  avoided. 

§  50.     Accuracy  of  Logarithmic  Results 

The  amount  of  $1.00  compounded  for  50  periods  at  3% 
is  seen  to  be  $4.383904.  The  result  is  slightly  inaccurate 
in  the  last  figure,  for  the  reason  that  two  decimal  places  were 
lost  by  multiplying.     Had  we  taken  the  ten-figure  logarithm 

on  page  XVIII  of  Chambers'  tables 0128372247 

this  multiplied  by  50  would  give 641861235 

or,  rounded  off  at  the  7th  place 6418612 

which  gives  the  more  accurate  result 4.383906 

§  51.     Logarithms  to  Fifteen  Places 

Since  it  is  necessary,  for  problems  involving  many 
periods,  to  use  a  very  extended  logarithm,  there  is  given  in 
Part  III  of  the  present  volume,  tables  of  fifteen-place  loga- 
rithms for  a  number  of  different  ratios  of  increase  (1  +i). 
These  are  at  much  closer  intervals  than  any  table  previously 
published,  and,  with  a  ten-figure  book  of  logarithms,  will 
give  exact  results  to  the  nearest  cent  on  $1,000,000.00. 

§  52.     Use  of  Logarithms  in  Present  Worth  Calculations 

We  will  further  exemplify  the  advantage  of  the  loga- 
rithmic method  by  solving  a  present  worth  problem.  Let 
it  be  required  to  find  the  present  worth  of  $1.00  due  in  50 
periods,  compounded  at  3%  per  period.  Multiplying  the 
logarithm  of  1.03  by  50,  just  as  in  §  50,  we  obtain 
.641861235.  But  it  is  the  reciprocal  of  1.03'^  or  1 -^ 
1.03^^  which  we  wish  to  obtain;  hence  we  must  subtract 
.641861235  from  the  logarithm  of  1,  which  is  0. 

0.000000000 
0.641861235 

Remainder,  1.358138765 


52  THE  MATHEMATICS  OF  INVESTMENT 

In  subtracting  a  greater  from  a  less  logarithm,  we  get  a 
negative  whole  number  (as  shown  by  the  minus  above),  the 
decimal  part  being  positive  and  obtained  by  ordinary 
subtraction. 

Neglecting  the  1,  for  the  moment,  we  search  in  the  right- 
hand  column  for  .358138765,  and  find  that  .3581253  is  the 
logarithm  of  2.2810;  and  proceeding  as  in  §50,  we  find 
that  .3581388  is  the  logarithm  of  2.281071.  The  decimal 
point,  however,  must  be  moved  one  place  to  the  left,  as 
directed  by  the  characteristic  1;  thus  giving  as  the  final 
result,  .2281071. 

By  means  of  multiplication,  we  may  check  the  results 
shown  in  this  and  the  foregoing  sections. 

By  §50,  1.03'Ms 4.383906 

As  above,  1  -^  1.03'"  is. . .     .2281071 

Since  these  two  results  are  reciprocals,  their  product  should 
equal  unity,  or  1.  The  result  of  the  multiplication  is 
1.0000000843326,  which  verifies  the  accuracy  of  the 
previous  computations. 


CHAPTER   IV 

AMOUNT   OF   AN    ANNUITY 

§  53.    Evaluation  of  a  Series  of  Payments 

We  have  now  investigated  the  two  fundamental  prob- 
lems in  compound  interest,  viz. :  to  find  the  amount  of  a 
present  worth,  and  to  find  the  present  worth  of  an  amount. 
The  next  question  is  a  more  complex  one:  to  find  the 
amount  and  the  present  worth  of  a  series  of  payments.  If 
these  payments  are  irregular  as  to  period,  value,  and  rate 
of  interest,  the  only  way  of  finding  the  amount  or  the  present 
worth  of  the  series  is  to  make  as  many  separate  computa- 
tions as  there  are  payments,  and  then  find  the  sum  of  the 
results  obtained.  But,  if  the  payments,  periods,  and  rates 
of  interest  are  uniform,  we  can  devise  a  method  for  finding 
by  one  operation  the  amount  or  present  worth  of  the 
whole  series. 

§  54.    Annuities 

A  series  of  payments  of  like  amounts,  made  at  regular 
periods,  is  called  an  annuity ;  the  period  does  not  necessarily 
need  to  be  a  year,  but  may  be  a  half-year,  a  quarter,  or  any 
other  length  of  time.  Thus,  if  an  agreement  is  made  pro- 
viding for  the  following  payments : 

September  9,  1914 $100.00 

March  9,  1915 100.00 

September  9,  1915 100.00 

March         9,  1916 100.00 

53 


54  THE  MATHEMATICS  OF  INVESTMENT 

there  would  be  an  annuity  of  $200.00  per  annum,  payable 
semi-annually;  or,  in  other  words,  an  annuity  of  $100.00  for 
each  half-year  period,  terminating  after  four  periods.  As- 
-suming  the  rate  of  interest  to  be  6%  per  annum,  payable 
semi-annually  (3%  per  period),  let  us  suppose  that  it  is 
required  to  find  the  total  amount  to  which  the  annuity  will 
have  accumulated  on  March  9,  1916,  and  the  present  worth, 
on  March  9,  1914,  of  this  series  of  future  payments.  It  is 
evident  that  the  answer  to  the  first  question  will  be  greater 
than  $400.00,  and  that  the  answer  to  the  second  question,  as 
shown  in  the  next  chapter,  will  be  less  than  $400.00. 

§  55.     Amount  of  Annuity 

It  is  easy,  in  this  case,  to  find  the  separate  amounts  of 
the  payments,  since  the  number  of  terms  is  very  small  and 
since  we  may  avail  ourselves  of  the  computations  in  §  30. 
A  schedule  could  be  made  as  follows : 

Date  of  Payment  Amount  at  March  9,  1916 
March          9,  1916  $100.00 

September   9,  1915  103.00 

March  9,  1915  106.09 

September  9,  1914  109.2727 

Total,  $418.3627 


§  56.     Calculation  of  Annuity  Amounts 

If,  however,  there  were  50  terms  instead  of  4,  the  work 
of  computing  these  50  separate  amounts,  by  the  use  of  loga- 
rithms, or  by  the  shorter  process  (in  this  case)  of  simple 
multiplication,  would  be  very  tedious.  To  shorten  the  process 
let  us  make  up  three  columns  of  amounts  for  four  periods, 
the  first  being  amounts  of  $1.00,  the  second  being  amounts 
of  $1.03,  and  the  third  being  amounts  of  $.03.  The  figures 
in  the  second  column  will  accordingly  be  1.03  times  the 


AMOUNT  OF  AN  ANNUITY 


55 


corresponding  figures  in  the  first  column,  while  the  figures 
in  the  third  column  will  be  the  difference  between  the  corre- 
sponding figures  in  the  first  two  columns. 


(1> 

Amounts 
of  $1.00 

(2) 

Amounts 

of  $1.03 

(3) 

Amounts 

of  $.03 

$1.00 
1.03 
1.0609 
1.092727 

Total,  $4.183627 

$1.03 
1.0609 
1.092727 
1.12550881 

$4.30913581 

$.03 
.0309 
.031827 
.03278181 

$.12550881 

§  57.    Formation  of  Tables 

We  may  take  the  diflFerence  between  the  totals  of 
columns  (1)  and  (2)  without  actually  finding  these  totals. 
It  will  be  observed  that  the  first  three  items  in  column  (2) 
are  the  same  as  the  last  three  items  of  column  (1).  The 
difference  between  the  totals  of  the  two  columns,  therefore, 
is  the  same  as  the  difference  between  the  last  item  of  (2)  and 
the  first  item  of  (1);  that  is,  $1.12550881  less  $1.00,  or 
$.12550881.  This  latter  figure  equals  the  total  of  column 
(3). 

§  58.    Use  of  Tables 

It  is  evident  that  an  annuity  of  three  cents  will  amount, 
under  the  conditions  assumed,  to  twelve  cents  and  the 
decimal  .550881.  Accordingly,  an  annuity  of  one  cent  will 
amount  to  one-third  of  $.12550881,  or  $.04183627.  An 
annuity  of  $1.00  will  amount  to  100  times  as  much,  or 
$4.183627,  while  an  annuity  of  $100.00  will  amount  to 
$418.3627,  which  agrees  exactly  with  the  result  obtained  by 
addition,  in  §  55. 


56  THE  MATHEMATICS  OF  INVESTMENT 

§  59.    Compound  Interest  as  a  Base  for  Annuity  Calculations 

The  amount  $.12550881  (obtained  by  subtracting  $1.00 
from  $1.12550881)  is  the  compound  interest  on  $1.00  for 
the  given  rate  and  time,  and  the  amount  $.03  is  the  single 
interest.  The  compound  interest  on  $1.00,  compounded 
semi-annually  at  6%,  up  to  any  time  corresponds  with  the 
amount  of  an  annuity  of  three  cents,  payable  on  exactly 
the  same  plan.  The  amount  of  the  annuity  of  $1.00  is 
$.12550881-^.03,  or  $4.183627;  and  from  this  we  formu- 
late the  rule  given  in  the  following  section. 

§  60.     Rule  and  Formula  for  Finding  Amount 

To  find  the  amount  of  an  annuity  of  $1.00  for  a  given 

time  and  at  a  given  rate,  divide  the  compound  interest  for 

the  total  number  of  periods,  by  the  single  interest  for  one 

period,  both  expressed  decimally. 

To  express  the  rule  in  a  formula,  let  A  represent  the 

amount,  not  of  a  single  $1.00,  but  of  an  annuity  of  $1.00; 

then  A  =  I  -T-  i. 

§  61.     Operation  of  Rule 

To  illustrate,  let  us  take  the  case  worked  out  in  §  50, 
where  we  found  the  amount  of  a  single  dollar  at  3%,  for  50 

periods,  to  be. $4.383906 

Subtracting  one  dollar 1.000000 

The  compound  interest  is , $3.383906 

Divide  this  by  .03  and  we  have $112.79687 

which  is  the  amount  to  which  50  payments  of  $1.00  each, 
at  3%  per  period,  would  accumulate. 


CHAPTER  V 

PRESENT  WORTH  OF  AN  ANNUITY 

§  62.    Method  of  Calculation 

To  find  the  present  worth  of  an  annuity,  we  can,  of 
course,  find  the  present  worth  of  each  payment,  and  then, 
by  addition,  find  the  total  present  worth  of  all  the  payments ; 
but  it  will  save  much  labor  if  we  compute  the  total  in  one 
operation,  as  we  computed  the  amount,  and  a  similar  course 
of  reasoning  will  lead  to  the  desired  result. 

§  63.    Tables  of  Present  Worth 

In  the  second  column  of  the  following  table  is  shown 
the  present  worth  of  $1.00  for  4,  3,  2  and  1  period,  respec- 
tively, at  3%  per  period ;  and  in  the  third  and  fourth  columns 
are  shown  similar  values  of  $1.03  and  $.03,  respectively. 


(1) 

Number 

of 
Periods 

(2) 
Present 
Worths 
of  $1.00 

(3) 
Present 
Worths 
of  $1.03 

(4) 
Present 
Worths 
of  $.03 

4 
3 
2 
1 

$.888487 
.915142 
.942596 
.970874 

$3.717099 

$.915142 
.942596 
.970874 

1.000000 

$.026655 
.027454 
.028278 
.029126 

Total, 

$3.828612 

$.111513 

57 


58  THE  MATHEMATICS  OF  INVESTMENT 

§  64.     Short   Method   for  Finding   Present   Worth   of   an 

Annuity 

Since  the  last  three  items  in  column  (2)  are  the  same 
as  the  first  three  items  in  column  (3),  it  is  evident  that,  in 
order  to  obtain  the  difference  between  the  totals  of  columns 
(2)  and  (3),  it  is  not  necessary  to  make  the  actual  additions 
of  these  columns,  but  merely  to  find  the  difference  between 
the  items  not  found  in  both  columns.  These  items  are  only 
two,  viz.,  $.888487  in  the  second  column,  and  $1.000000  in 
the  third  column.  Their  difference  is  $.111513,  which  agrees 
with  the  total  found  by  the  addition  of  column  (4). 

§  65.     Present  Worth  Obtained 

The  difference  between  the  $.888487  of  the  second 
column  and  $1.000000  of  the  third  column,  amounting  to 
$.111513,  is  the  compound  discount  of  $1.00  for  four 
periods  at  3%.  When  this  difference  is  divided  by  the  single 
interest  (.03),  we  obtain  $3.71710,  which  is  the  same  result 
(rounded  up)  as  that  obtained  by  adding  column  (2).  From 
this  observation,  we  construct  the  rule  given  in  the  following 
section : 

§  66.     Rule  for  Present  Worth 

To  find  the  present  worth  of  an  annuity  of  $1.00  for  a 
given  time  at  a  given  rate,  divide  the  compound  discount  for 
that  time  and  rate  by  the  single  interest  for  one  period,  both 
expressed  decimally. 

§  67.    Formulas  for  Present  Worth 

In  symbols,  the  rule  may  be  expressed,  P  =  D  -^  /.  Since, 
by  §  35,  D  =  I  -^ a,  we  obtain  F  =  l-^a-^i,  or  P  =  I-^ 
i  -f-  a.  And  since,  by  §  60,  A  =  I  -^-  i,  there  comes  the  re- 
sulting symbolic  rule,  P  =  A  -^-  a,  the  latter  part  of  this 
equation  signifying  the  present  worth  of  the  amount  of  the 


PRESENT  WORTH  OF  AN  ANNUITY  59 

annuity.    Summarizing,  therefore,  we  have  the  two  symbolic 
rules : 

P  =  A-f-a 

§  68.     Analysis  of  Annuity  Payments 

It  may  assist  in  acquiring  a  clear  idea  of  the  working  of 
an  annuity,  if  an  analysis  is  given  of  a  series  of  annuity  pay- 
ments from  the  point  of  view  of  the  purchaser.  For  this 
purpose  we  will  suppose  that  a  person  investing  $3.7171 
at  3%,  in  an  annuity  of  $1.00  per  period,  payable  at  the 
end  of  each  period,  expects  to  receive  at  each  payment,  be- 
sides 3%  on  his  principal  to  date,  a  portion  of  that  principal, 
and  thus  to  have  his  entire  principal  gradually  repaid. 

His  original  principal  is , $3.7171 

At  the  end  of  the  first  period,  he  receives : 

3%  on  $3.7171 $.1115 

Payment  on  principal 8886         .8885 

Total $1.0000 

Leaving  new  principal  (which  is  equiva- 
lent to  the  present  worth  at  three  periods)     $2.8286 

At  the  end  of  the  second  period,  he  receives : 

3%   on  $2.8286 $.0849 

Payment  on  principal ,..,..        .9151         .9151 

Total $1.0000 

Leaving  new  principal $1.9135 

At  the  end  of  the  third  period,  he  receives : 

3%   on  $1.9135 $.0574 

Payment  on  principal 9426         .9426 

Total $1.0000 

Leaving  new  principal $.9709 


6o  THE  MATHEMATICS  OF  INVESTMENT 

At  the  end  of  the  last  period,  he  receives : 

3%  on  $.9709 $.0291 

Payment  on  principal  in  full 9709         .9709 

Total $1.0000 

In  the  above  manner  we  find  that  the  annuitant  has  re- 
ceived interest  in  full  on  the  principal  outstanding,  and  has 
also  received  the  entire  original  principal.  The  correctness 
of  the  basis  on  which  we  have  been  working  is  thus 
corroborated. 

§  69.     Components  of  Annuity  Instalments 

It  is  usual  to  form  a  schedule  showing  the  components 
of  each  instalment  in  tabular  form : 


Date 

Total 
Payment 

Payments 

of 

Interest 

Payments 

on 
Principal 

Principal 

Out- 
standing 

March        9,  1914. 
September  9,  1914. 
March         9,  1915. 
September  9,  1915. 
March        9,  1916. 

$1.00 
1.00 
1.00 
1.00 

$4.00 

$.1115 
.0849 
.0574 
.0291 

$.2829 

$.8885 
.9151 
.9426 
.9709 

$3.7171 

$3.7171 
2.8286 
1.9135 
0.9709 
0.0000 

Had  the  purchaser  reinvested  each  instalment  at  3%,  he 
would  have,  at  the  end,  $4.1836  (§55),  which  is  equivalent 
to  his  original  investment  compounded  ($3.7171  X  1.1255  = 
$4.1836). 

§  70.    Amortization 

The  payments  on  principal  are  known  as  amortization, 
which  may  be  defined  as  the  gradual  repayment  of  a  principal 
sum  through  the  resultant  operation  of  two  opposing  forces 
— ^periodical  payments  and  compound  interest.    The  effect 


PRESENT  WORTH  OF  AN  ANNUITY  6l 

of  the  periodical  payments  is  to  reduce  the  principal  sum, 
while  the  effect  of  the  compound  interest  is  to  increase  it. 
In  ordinary  compound  interest,  each  new  principal  is  greater 
than  the  preceding  principal ;  while  in  the  case  of  amortiza- 
tion, each  principal  is  less  than  the  preceding  one. 

§  71.    Amortization  and  Present  Worth 

It  will  be  noticed,  from  §  69,  that  each  payment  on 
principal,  or  amortization,  for  one  period,  is  the  present 
worth  of  the  instalment  at  the  beginning  of  its  period.  For 
example,  at  the  end  of  the  first  period,  September  9,  1914,  a 
payment  on  principal  is  made  amounting  to  $.8885,  which 
is  the  present  worth  of  the  instalment  paid  on  that  date 
($1.00)  for  four  periods  at  3%.  From  this  fact,  it  follows 
that,  if  we  know  the  amount  of  the  instalment,  the  rate,  and 
the  number  of  remaining  periods,  we  can  calculate  the 
amortization  included  in  the  instalment. 

§  72.     Development  of  a  Series  of  Amortizations 

It  will  also  be  noticed  that  each  amortization  multiplied 
by  1.03  becomes  the  next  following,  these  being  a  series  of 
present  worths ;  and  that  thus  they  may  be  derived  from  one 
another,  upwards  or  downwards. 

§  73.     Evaluation  by  Logarithms 

In  §  52,  by  the  use  of  logarithms,  we  found  the  present 
worth  of  $1.00  for  50  periods,  at  3%,  to  be. .  $.2281071 
Subtracting  this  from 1.0000000 

we  have  the  compound  discount ,. ., ,       $.7718929 

Dividing  this  by  .03,  we  have $25.72976  + 

which  is  the  present  worth  of  an  annuity  of  $1.00  for  50 
periods,  at  3%.  Thus  we  see  that  the  process  of  finding  the 
present  worth  of  an  annuity,  or,  as  it  is  termed,  evaluation,  is 
rendered  easy — no  matter  how  long  the  time — by  using 
logarithms. 


CHAPTER  VI 

SPECIAL  FORMS  OF  ANNUITIES 

§  74.     Ordinary  or  Immediate  Annuities 

The  annuities  heretofore  spoken  of  are  payable  at  the 
end  of  each  period,  and  are  the  kind  most  frequently  occur- 
ring. To  distinguish  them  from  other  varieties,  they  are 
spoken  of  as  ordinary  or  immediate  annuities. 

§  75.     Annuities  Due 

When  the  instalments  of  an  annuity  are  payable  at  the 
beginning  of  their  respective  periods,  the  annuity  is  called 
an  annuity  due,  although  prepaid  would  seem  more  natural. 
It  is  evident  that  this  is  merely  a  question  of  dating.  The 
instalments  compared  with  those  in  §  56  are  as  follows : 


Immediate 

Annuity 

Immediate 

Annuity 
4  Periods 

Due 
4  Periods 

Annuity 
5  Periods 

r 

$1.00 

$1.03 

$1.00 

Amounts  of 

1.03 
1.0609 

1.0609 
1.0927 

1.03 

1.0609 

$1.00 

1.0927 

1.1255 

1.0927 

1.1255 

$5.3091 

—  1.0000 

$4.1836 

$4.3091 

$4.3091 

62 


SPECIAL   FORMS    OF   ANNUITIES 


63 


Hence,  to  find  the  amount  of  an  annuity  due,  for  any 
number  of  periods,  say  t  periods,  find  the  amount  of  an 
immediate  annuity  for  ^  + 1  periods,  and  subtract  one 
instalment. 

§  76.    Present  Worth  of  Annuities  Due 

In  regard  to  present  worths,  the  instalments  compared 
with  those  in  §  63  would  be  as  follows : 


Immediate 

Annuity 
4  Periods 

Annuity- 
Due 
4  Periods 

Immediate 

Annuity 

3  Periods 

Present 
Worths  of    - 
$1.00 

$.888487 
.915142 
.942596 
.970874 

$.915142 
.942596 
.970874 

1.000000 

$.915142 
.942596 

.970874 

$2.828612 
+  1.000000 

$3.717099 

$3.828612 

$3.828612 

Therefore,  to  find  the  present  worth  of  an  annuity  dtw 
for  /  periods,  find  the  present  worth  of  an  immediate  annuity 
for  ;  —  1  periods,  and  add  one  instalment. 

§  77.    Present  Worth  of  Deferred  Annuities 

A  deferred  annuity  is  one  which  does  not  commence  to 
run  immediately,  but  only  after  a  certain  number  of  periods 
have  elapsed.  Thus,  an  annuity  of  5  terms,  4  terms  deferred, 
would  commence  at  the  beginning  of  the  fifth  period,  and 
continue  to  the  end  of  the  ninth  period. 

If  there  were  nine  terms  in  the  annuity,  none  being  de- 
ferred, and  if  the  ratio  of  increase  were  assumed  to  be  r  and 


64  THE  MATHEMATICS  OF  INVESTMENT 

the  present  worth  of  the  first  term  were  assumed  to  be 
unity,  the  present  worth  of  the  annuity  for  nine  terms 
would  be : 

l+7  +  ^  +  ;;:i  +  ;:5  +  J  +  ^  +  J  +  ^(§§  18,  66) 
The  present  worth  of  the  annuity  for  four  terms  would  be : 

The  present  worth  of  the  annuity  for  the  five  deferred  terms 
would,  of  course,  be  the  difference  between  the  above  two 
sums,  or: 

i+l+i.i.i 

^4  ~  ^5  ~  ^6     I     ^7    I    ^8 

§  78.  Rule  for  Finding  Present  Worth  of  Deferred  Annuity 
From  the  foregoing,  we  derive  the  rule:  To  find  the 
present  worth  of  an  annuity  for  m  terms,  deferred  for  n 
terms,  subtract  the  present  worth  of  an  annuity  for  n  terms 
from  the  present  worth  of  an  annuity  for  m-\-  n  terms. 

§  79.     Present  Worth  of  Perpetuities 

A  perpetual  annuity,  or  a  perpetuity,  is  one  which  never 
terminates.  Its  amount  is  infinity,  but  its  present  worth  can 
be  calculated  at  any  given  rate  of  interest.  If  each  instalment 
of  an  annuity  is  $1.00*  and  the  rate  5%,  the  value  of  the 
annuity  is  such  a  sum  as  will  produce  $1.00  at  that  rate. 
This  sum  is  $20.00,  being  $1.00  -f-  6%.  The  compound  dis- 
count is  the  entire  $1.00,  being  for  an  infinite  number  of 
terms.  Therefore,  the  rule  of  §  66  still  holds  true :  divide 
the  compound  discount  by  the  single  rate  of  interest,  in  order 
to  find  the  present  worth  of  the  annuity. 


SPECIAL  FORMS  OF  ANNUITIES  65 

§  80.    Perpetuity  in  Stock  Purchased  for  Investment 

A  share  of  stock  may  be  treated  in  the  same  manner  as  a 
perpetuity,  provided  its  dividend  is  assumed  to  continue  at 
a  fixed  rate.  If  the  dividend  is  $4.00  per  share,  and  if  it  is 
desired  to  purchase  at  such  a  basis  as  to  yield  6%  on  the 
investment,  the  price  per  share  should  be  $4.00-^-6%,  which 
equals  $66.67.  This  price  is  irrespective  of  the  nominal  or 
par  value  of  the  stock.  Both  in  perpetuities  and  in  shares  of 
stock,  the  price  =  c-^i. 

§81.    When  Annuity  Periods  and  Interest  Periods  Differ 

In  all  of  these  examples  of  annuities,  it  has  been  assumed 
that  the  term  or  interval  between  payments  is  the  same 
length  of  time  as  the  interest  period.  It  frequently  happens, 
however,  that  the  rate  of  interest  is  stated  to  be  so  much  per 
year,  while  the  payments  are  half-yearly  or  quarterly;  or 
there  may  be  yearly  payments,  while  the  desired  interest 
rate  is  to  be  on  a  half-yearly  basis.  We  shall  defer  the  treat- 
ment of  these  latter  cases  until  the  subject  of  nominal  and 
effective  rates  of  interest  has  been  discussed. 

§  82.     Varying  Annuities 

There  may  also  be  varying  annuities,  where  the  instal- 
ment changes  by  some  uniform  law.  These  seldom  occur 
in  practice.  Where  the  change  is  simple,  as  in  arithmetical 
progression,  the  total  annuity  may  be  regarded  as  the  sum 
of  several  partial  annuities;  otherwise  the  values  must  be 
separately  calculated  for  each  term.  An  annuity  running 
for  five  terms,  as  follows :  13,  18,  23,  28,  33,  may  be  re- 
garded as  the  sum  of  the  following : 

(1)  an  annuity  of  13  for  6  terms; 

(2)  an  annuity  of  5  for  4  terms; 


66  THE  MATHEMATICS  OF  INVESTMENT 

(3)  an  annuity  of  5  for  3  terms; 

(4)  an  annuity  of  5  for  2  terms;  and 
;(5)  an  annuity  of  5  for  1  term. 

In  actual  practice,  in  a  case  of  this  kind,  in  order  to  find 
the  amount  or  the  present  worth  of  the  annuity,  it  would 
probably  be  easiest  to  find  the  amount  or  the  present  worth 
of  each  term,  and  then  find  the  total  of  these  separate  items. 


CHAPTER  VII 
RENT  OF  ANNUITY  AND  SINKING  FUND 

§  83.     Rent  of  Annuity 

The  number  of  dollars  in  each  separate  payment  of  an 
annuity  is  called  the  rent  of  the  annuity. 

In  §  63,  we  saw  that  $3.Y171  is  the  present  worth,  at 
3%,  of  an  annuity  composed  of  4  payments  of  $1.00  each. 
We  may  reverse  this  and  say  that  $1.00  is  the  rent  of 
$3.7171  invested  in  an  annuity  of  4  payments  at  3%.  What, 
then,  is  the  rent  to  be  obtained  by  investing  $1.00  in  the 
same  way?  Since  the  present  worth  has  been  reduced  in 
the  ratio  of  3.7171  to  1,  evidently  the  rent  must  be  reduced 
in  the  same  ratio,  that  is,  1  -^  3.7171.  By  ordinary  division 
or  by  logarithms,  this  quotient  is  .26903.  Therefore, 
$.26903  is  the  rent  of  an  annuity  of  4  terms  at  3%,  for 
every  $1.00  invested;  or  $1.00  is  the  present  worth  at  3% 
of  an  annuity  for  4  years  of  $.26903.  This  may  be  illus- 
trated by  making  up  a  schedule : 


Rent 

Interest 

Reduction 
or  Amorti- 
zation 

Value 

Beginning  of  first  period. 

End  of  first  period 

End  of  second  period 

End  of  third  period 

End  of  fourth  period 

$  .26903 
.26903 
.26903 
.26903 

$.03 
.02283 

.01544 
.00785 

$  .23903 
.24620 
.25359 
.26118 

$1.00000 

.76097 

.51477 

.26118 

0. 

$1.07612 

$.07612 

$1.00000 

67 


(^  THE  MATHEMATICS  OF  INVESTMENT 

§  84.     Rule  for  Finding  Rent  of  Annuity 

To  find  the  rent  of  an  annuity  valued  at  $1.00,  divide 
$1.00  by  the  present  worth  of  an  annuity  of  $1.00  for  the 
given  rate  and  time.  Rent  =  1  ->  P ;  and  since,  by  §  67, 
P  =  D ^ i^  and  F  =  A-7-  a,  we  obtain  two  other  symbolic 
rules : 

Rent  =  i-^D 

Rent  =  a  -^-  A 

§  85.    Alternative  Method  of  Finding  Rent 

An  alternative  method  of  determining  the  value  of  the 
rent  of  an  annuity  is  to  form  a  proportion,  as  in  arithmetic, 
and  then  solve  the  proportion.    For  example : 

Rent  of  Annuity         Present  Worth  of  Annuity 
$1.00     :    X     ::  $3.7171     :     $1.00 

In  other  words,  if  a  rent  of  $1.00  produces  a  present  worth 
of  $3.7171,  then  what  quantity  of  rent  will  produce  a  present 
worth  of  $1.00?  Multiplying  the  two  extremes  together, 
and  dividing  the  product  by  the  mean,  we  find  the  other 
mean  to  be  $.26903,  which  is  the  rent  required. 

§  86.     Rent  of  Deferred  Payments 

The  problem  of  finding  the  rent  of  an  annuity  may  be 
regarded  as  equivalent  to  another  problem — that  of  finding 
how  much  per  period  for  n  periods,  at  the  rate  i,  can  be 
bought  for  $1.00.  A  borrower  may  agree  to  pay  back  a  loan 
in  instalments,  each  of  which  comprises  both  principal  and 
interest.  Suppose  that  a  loan  of  $1,000  were  made  under 
the  agreement  that  such  a  uniform  sum  should  be  paid 
annually  as  would  pay  off  (amortize)  the  entire  debt  with 
3%  interest  in  4  years.  The  present  worth  is,  of  course, 
$1,000,  and  by  the  above  process  each  instalment  or  con- 
tribution would  be  $269.03.    In  countries  imposing  an  in- 


RENT  OF  ANNUITY  AND  SINKING  FUND 


69 


come  tax,  it  is  usual  to  incorporate  in  agreements  of  this 
nature  a  schedule  showing  what  part  of  the  instalment  is 
interest — since  that  alone  is  taxable — somewhat  as  follows : 


Annual 
Instalment 

[nterest  on 
Balance 

Payment  on 
Principal 

Principal 
Outstanding 

January       1, 1914 
December  31, 1914 
December  31, 1915 
December  31, 1916 
December  31, 1917 

$269.03 
269.03 
269.03 
269.03 

$30.00 
22.83 

15.44 

7.85 

$239.03 
246.20 
253.59 
261.18 

$1,000.00 

760.97 

514.77 

261.18 

0. 

$1,076.12 

$76.12 

$1,000.00 

§  87.    Annuities  as  Sinking  Funds 

One  other  question  arises  with  regard  to  annuities,  and 
that  is  in  the  cnse  of  an  annuity  so  constructed  as  to  accumu- 
late to  a  certain  amount  at  a  certain  time.  The  amount  to 
be  accumulated  is  called  a  sinking  fund.  Frequently  the 
uniform  periodical  contribution  is  itself  called  the  sinking 
fund,  but,  more  strictly  speaking,  it  should  be  called  the 
sinking  fund  contribution. 

In  the  case  exhibited  in  the  schedule  of  §  86,  the  debt 
was  amortized,  with  the  assent  of  the  creditor,  by  gradual 
payments.  Let  us  suppose,  however,  that  the  creditor  pre- 
fers to  wait  until  the  day  of  maturity,  and  receive  his  $1,000 
all  at  one  time,  instead  of  by  partial  payments.  The  debtor 
must  pay  interest  amounting  to  $30.00  each  year,  but,  in 
addition  to  this,  in  order  to  provide  for  the  principal  on  a 
sinking  fund  plan,  he  must  transfer  from  his  general  assets 
to  a  special  account  (or  into  the  hands  of  a  trustee)  such 
an  annual  sum  as  will  accumulate,  in  4  years  at  3%,  to 
$1,000.     Since  $1.00,  set  aside  annually,  amounts,  after  4 


70 


THE  MATHEMATICS  OF  INVESTMENT 


years  on  a  3%  basis,  to  $4.183627  (§56),  to  find  what 
sum  will  similarly  amount  to  $1,000,  we  must  divide  1,000 
by  4.183627.  In  this  manner  the  sinking  fund  contribution 
is  found  to  be  $239.03. 

§  88.     Rule  for  Finding  Sinking  Fund  Contributions 

To  find  what  annuity  will  amount  to  $1.00,  or  what 
should  be  each  sinking  fund  contribution  to  provide  for 
$1.00:  divide  $1.00  by  the  amount  of  an  annuity  of  $1.00 
for  the  given  rate  and  time.  In  symbols,  sinking  fund  con- 
tribution, or  S.  F.  C,  =  1  -^  A ;  or  (since  A  =  I  ^i,  per 
§  60)  it  also  equals  i  -h  I. 

Put  in  the  form  of  a  proportion,  the  question  of  §  87 
would  appear  as  follows : 


Sinking  Fund  Contribution 
$1.00      :     X  :: 


Sinking  Fund 
$4.183627      :     $1,000.00 


The  unknown  quantity,  x,  would  be  the  same  as  before, 
$239.03. 

§  89.     Verification  Schedule 

The  correctness  of  the  result  found  in  §  87  may  be  proved 
by  a  schedule  constructed  in  the  following  manner : 


Annual  Sink- 
ing Fund 
Contribution 

Interest  Dur- 
ing Preced- 
ing Year 

Total 
Addition 
to  Sinking 
Fund 

TotalAmount 
Accumulated 
in  Sinking 
Fund 

January       1,  1914 
December  31,  1914 
December  31,  1915 
December  31,  1910 
December  31,  1917 

$239.03 
239.03 
239.03 
239.03 

$  7.17 
14.56 
22.15 

$239.03 
246.20 
253.59 
261.18 

$     0. 
239.03 
485.23 
738.82 

1,000.00 

$056.12 

$43.88 

$1,000.00 

RENT  OF  ANNUITY  AND  SINKING  FUND 


71 


§  90.     Amortization  and  Sinking  Fund 

On  comparing  the  schedules  in  §§  86  and  89,  we  find 
that  the  annual  instalments  or  contributions  are  respectively 
$269.03  and  $239.03,  the  difference  of  which  is  $30.00,  or 
exactly  the  yearly  interest  on  the  original  loan  of  $1,000.00. 
Hence,  the  amount  paid  in  the  second  case,  if  interest  be 
included,  is  just  the  same  as  in  the  first  case.  Gradual  pay- 
ments on  account  of  a  debt,  or  gradual  accumulations  hav- 
ing in  view  one  single  final  payment  in  full,  therefore  amount 
to  the  same  thing. 

As  a  provision  for  liquidating  indebtedness,  or  for  re- 
placing vanishing  assets,  sinking  fund  and  amortization  are 
two  different  applications  of  the  same  principle.  Formerly, 
the  terms  were  used  interchangeably,  but  more  recently  they 
are  distinguished  as  follows : 

(1)  The  sinking  fund  method  permits  the  debt  to  stand 
until  maturity,  but  in  the  meantime  accumulates  a  fund 
which  at  maturity  pays  off  the  entire  debt,  the  interest  on 
the  original  sum  being  paid  separately. 

(2)  The  amortization  method  accumulates  nothing,  but 
gradually  reduces  the  debt,  the  amount  of  the  reduction 
being  the  excess  of  the  periodical  payment  over  the 
periodical  interest. 


CHAPTER  VIII 

NOMINAL    AND    EFFECTIVE    RATES 

§  91.     Explanation  of  Terms 

In  the  previous  chapters,  all  of  our  computations  re- 
garding interest  have  been  based  upon  a  certain  number  of 
periods  and  upon  a  certain  rate  per  period.  In  the  business 
world,  it  is  usual  to  speak  of  interest  rates  as  so  much  per 
annum.  In  the  vast  majority  of  instances,  however,  the  in- 
terest, although  it  is  either  designated  or  understood  to  be 
per  annum,  is,  nevertheless,  not  paid  by  the  year  (that  is, 
once  a  year),  but  in  semi-annual  or  quarterly  instalments. 
Where  the  interest  is  payable  otherwise  than  annually,  the 
rate  per  annum  is  only  nominally  correct.  For  example,  if 
on  May  1, 1914,  we  lend  $1,000.00  at  6%,  interest  to  be  paid 
semi-annually,  the  interest  account  for  the  year  would  be 
as  follows : 

November  1,  1914,  Interest  earned. . ., ,. .   $30.00 

May  1,  1915,  Interest  earned: 

On  original  loan , 30.00 

On  the  $30.00  received  on  November  1,  1914, 

for  6  months  at  an  assumed  rate  of  6% ... .         .90 

Total $60.90 


The  total  interest  earnings  during  the  year,  therefore, 
would  be  $60.90,  which  is  at  the  effective  rate  of  6.09% 
on  the  original  investment,  as  compared  with  a  nominal 
rate  of  6%. 

72 


fjj 


NOMINAL  AND  EFFECTIVE  RATES 


73 


§  92.     Semi-Annual  and  Quarterly  Conversions 

In  the  example  given  in  the  preceding  section,  the  in- 
terest is  payable  (or,  as  it  is  frequently  called,  convertible) 
semi-annually.  The  true  or  effective  rate  for  each  half- 
yearly  period  is  therefore  3%,  and  the  ratio  of  increase  is 
1.03.  The  amount  at  the  end  of  the  year  would  be  the 
square  of  1.03,  or  1.0609,  thus  giving  6.09%  as  the  effective 
rate  per  annum.  In  the  case  of  quarterly  conversion,  the 
amount  at  the  end  of  the  year  v^ould  be  the  fourth  power  of 
1.015,  or  1.061364,  giving  6.1364=%  as  the  effective  annual 
rate.  The  following  table  shows  the  effective  annual  rates 
for  various  periods  of  conversion,  the  nominal  annual  rate 
being  6%  : 


Period  of                                                  ^^      .  .           ,  r>  . 

^             .                                                    Effective  Annual  Rate 
Conversion 

Yearly 1.06  —  1  or  6.0000% 

Semi-annually 1.03'  —  1  or  6.0900% 

Quarterly 1.015*  —  1  or  6.1364% 

Monthly , 1.005'^  —  1  or  6.1678%? 


Y        365/ 


Daily f  1+-^  I      —1     or  6.1826% 


§  93.     Limit  of  Effective  Annual  Rate 

It  will  be  seen  that  the  effective  rate  increases  as  the 
conversions  become  more  frequent.  There  is  a  limit,  how- 
ever, beyond  which  this  acceleration  will  not  go.  If  an  in- 
vestment on  a  6%  nominal  annual  rate  is  compounded  every 
minute,  or  every  second,  or  every  millionth  of  a  second,  or 
constantly,  the  effective  annual  rate  could  never  be  so  great 
as  6.184%.* 


'See  §   238. 


74 


THE  MATHEMATICS  OF  INVESTMENT 


§  94.     Rule  for  Effective  Rate 

From  observation  of  the  table  shown  in  §  92,  we  may 
deduce  the  following  symbolic  rule  for  finding  the  effective 
rate,  m  representing  the  number  of  payments  per  annum, 
and  y  the  effective  rate : 


=(i+i)'"- 


§  95.     Logarithmic  Process 

In  order  to  exemplify  logarithmic  processes  in  working 
out  the  foregoing  rule,  let  it  be  required  to  find  the  effective 
rate  of  interest  when  the  nominal  rate  is  6%  per  annum, 
compounded  daily.    The  rule  in  §  94  then  becomes : 

By  the  use  of  logarithms,  we  obtain : 

log.  .06  =2.7Y81513 

log.  365  =2.5622929 

Hence,  log.   (.06 -^  365)  =4.2158584 

4.2158584  is,  we  find,  the  logarithm  of  .0001643835;  and, 
therefore,  the  value  found  thus  far  is : 

y=(l +  .0001643835)  ^^^  —  1 
or,  ;  =  1.0001643835'«'  — 1 

The  logarithm  of  1.0001643835  is  .00007138 ;  and  365  times 
this  latter  figure  is  .02605370,  which  we  find  to  be  the  loga- 
rithm of  1.061826.  The  value  for  the  effective  rate  then 
becomes : 

y  =  1.061826  —  1 
or,  y=    .061826,  or  6.1826% 


CHAPTER   IX 

BONDS  AND  THE  PROPER  BASIS  OF  BOND 
ACCOUNTS 

§  96.     Provisions  of  Bonds 

The  most  common  forms  of  interest-bearing  securities 
are  bonds.    Every  bond  contains  a  complex  promise  to  pay : 

(1)  A  certain  sum  of  money  at  a  stipulated  future  time, 

this  sum  being  known  as  the  principal,  or  par. 

(2)  Certain  smaller  sums,  proportionate  to  the  principal, 

and  payable  at  various  earlier  times  than  the 
principal. 

These  smaller  sums  are  usually  known  as  the  interest 
payments,  but,  as  they  do  not  necessarily  correspond  to  the 
true  rate  of  interest,  it  will  be  better  to  speak  of  them  as  the 
coupons. 

Bonds  also  contain  provisions  as  to  the  time,  place,  and 
manner  of  these  payments,  and  usually  refer,  also,  to  the 
mortgage,  if  any,  made  to  insure  their  fulfillment,  and  to  the 
law,  if  any,  authorizing  the  issue. 

§97.     Interest  on  Bonds 

The  rate  of  interest  named  in  a  bond  is  usually  an  integer 
per  cent,  or  midway  between  two  integers:  as,  2%,  2%%, 
3%,  31/2%,  4%,  41/2%,  5%,  6%,  7%,  etc.  Occasionally 
such  odd  rates  occur  as  3%%,  3.60%,  3.65%,  33/4%,  but 
these  are  unusual  and  inconvenient.  Most  bonds  provide  for 
semi-annual  payments  of  interest ;  a  considerable  number  of 

75 


^6  THE  MATHEMATICS  OF  INVESTMENT 

issues,  however,  pay  interest  quarterly,  and  a  very  few 
annually.  With-  most  bonds,  the  interest  is  payable  on  the 
first  day  of  the  month.  In  the  case  of  a  very  few  bonds 
the  interest  falls  due  on-  the  15th  or  on  the  last  day  of  the 
month.  In  some  respects  it  would  be  better  if  bond  interest 
were  payable  on  the  last  day  of  a  calendar  month,  instead 
of  on  the  first  day  of  the  succeeding  month,  since  the  entire 
transaction  (including*  the  payment  of  cash  for  the  accrued 
interest)  would  thus  be  brought  inside  of  a  calendar  period. 
The  item  of  "Interest  Accrued"  on  monthly  balance  sheets 
would  in  this  manner  frequently  be  eliminated,  or  at  least 
substantially  reduced. 

§  98.     How  Bonds  Are  Designated 

Bonds  are  usually  designated  according  to  the  obligor, 
the  rate  of  interest,  the  date  of  maturity,  and  sometimes  the 
initials  of  the  months  when  interest  is  payable.  Thus,  "Man- 
hattan 4's  of  1990,  J  J"  indicates  the  bonds  of  the  Manhat- 
tan Railway  Company,  bearing  4%  interest  per  annum,  the 
principal  being  due  in  1990,  and  the  interest  coupons  being 
payable  semi-annually  in  January  and  July. 

§  gg.     Relation  of  Cost  to  Net  Income 

Bonds  are  seldom  bought  or  sold  at  their  exact  par 
value,  and  this  fact  has  an  effect  on  the  rate  of  net  income 
derived  from  the  original  investment.  If  the  amount  in- 
vested is  greater  than  the  par  value,  the  difference  is  known 
as  the  premium.  This  premium  is  not  repaid  at  maturity, 
as  is  the  par  value  or  principal  of  the  bond,  and  hence  must 
be  provided  for  out  of  the  various  interest  payments.  Thus, 
a.  bond  purchased  above  par  produces  a  lower  rate  of  in- 
come than  the  rate  of  interest  represented  by  the  coupons. 
Conversely,  if  the  purchase  is  below  par,  the  investor  will, 
at  maturity,  receive  not  only  the  amount  of  his  original  in- 


BONDS  AND  BOND  ACCOUNTS         77 

vestment,  but  also  the  difference  between  this  amount  and 
the  par  value  of  the  bond.  This  difference,  technically 
known  as  the  discount,  has  the  effect  of  making  the  rate  of 
income  higher  than  the  rate  of  interest  shown  by  the 
coupons. 

§  100.    Coupon  and  Effective  Rate  of  Interest  on  Bonds 

The  following  are  some  of  the  expressions  used  to  de- 
note an  investment  made  above  par :  "6%  bond  to  net  5%"; 
"6%  bond  on  5%  basis";  "6%  bond  yielding  5%";  "6% 
bond  paying  5%";  etc.  In  the  cases  of  bonds  bought  below 
par,  the  income  rate  would  be  larger  than  the  coupon  rate, 
as,  for  example,  "3%  bond  to  net  4%,"  etc.  In  all  of  the 
above  instances,  the  percentage  immediately  preceding  the 
word  "bond"  signifies  the  coupon  rate  of  interest,  while  the 
other  percentage  signifies  the  true  or  effective  rate  of 
interest. 

§  loi.    Present  Worth  of  Bonds 

It  will  be  seen,  therefore,  that  the  sale  of  a  bond  involves 
the  transfer  of  the  right  to  receive,  at  the  stipulated  times, 
both  the  principal  and  the  periodical  amounts  of  interest. 
None  of  these  various  sums  is  ever  worth  its  face  value,  or 
par,  until  the  arrival  of  its  stipulated  date  of  payment.  The 
principal  is  never  worth  its  face  value  until  its  maturity, 
and  the  coupons  are  never  worth  their  face  values  until 
their  respective  maturities.  Yet,  while  both  principal  and 
coupons  are  always  at  a  discount,  except  at  their  respective 
dates  of  maturity,  the  aggregate  value  or  present  worth  of 
the  principal  and  coupons  at  any  one  time  prior  to  maturity 
is  frequently  more  than  the  par  value  of  the  principal  alone 
(as  in  the  case  of  a  bond  bought  at  a  premium) ;  and  it  is 
this  aggregate  present  worth  of  both  principal  and  coupons 
which  is  always  the  question  at  issue  in  connection  with  the 
purchases  and  sales  of  bonds. 


y8  THE  MATHEMATICS  OF  INVESTMENT 

§  102.    Considerations  in  the  Purchase  of  Bonds 

In  fixing  the  price  which  he  is  wiUing  to  pay,  the  pur- 
chaser is  guided  by  several  considerations,  among  them  the 
following : 

(1)  The  amount  of  the  principal. 

(2)  The  date  of  maturity  of  the  principal. 

(3)  The  amount  of  each  coupon. 

(4)  The  number  of  coupons. 

(5)  The  dates  of  maturity  of  the  various  coupons. 

(6)  The  rate  of  interest  which  can  be  earned  upon 

securities  of  a  similar  grade. 

This  last  point  also  involves  a  determination  of  the  de- 
gree of  probability  that  the  principal  and  the  various  coupons 
will  be  promptly  paid  at  their  dates  of  maturity;  or,  in 
other  words,  consideration  must  be  given  to  the  financial 
reputation  and  integrity  of  the  obligor. 

§  103.    Present  Worth  and  Earning  Capacity  of  Bonds 

In  effect,  the  purchaser  of  a  bond  discounts,  at  a  certain 
fixed  rate,  the  principal  and  each  coupon  at  compound  inter- 
est, for  the  periods  which  they  respectively  have  to  run,  and 
the  sum  of  these  partial  present  worths  is  the  value  o'f  the 
bond.  If  he  can  buy  at  a  price  below  this  value,  he  will  re- 
ceive a  higher  rate  of  Interest  than  he  anticipated.  If  he  has 
to  pay  more  than  this  value,  his  rate  of  Interest  will  be  lower. 
As  he  cashes  each  coupon,  he  receives  what  he  paid  for  it, 
plus  compound  interest  at  the  uniform  rate;  thenceforward 
he  earns  interest  on  a  diminished  Investment  as  far  as  cou- 
pons are  concerned,  but  on  an  increased  investment  as  to 
principal.  If  the  par  value  of  his  coupons  Is  less  than  the 
total  interest  earned  during  the  period,  there  Is  an  Increase 
in  the  total  Investment;  if  such  par  value  is  greater,  then 
there  Is  a  surplus  which  operates  to  reduce  the  investment  or 
to  amortize  the  premium. 


BONDS  AND  BOND  ACCOUNTS  79 

§  104.     Cost  and  Par  of  Bonds 

There  are,  therefore,  two  fixed  points  in  the  history  of  a 
bond :  the  original  cost,  or  money  invested,  and  the  principal, 
or  par — the  money  to  be  received  at  maturity.  Between 
these  two  points  there  is  a  gradual  change :  if  bought  below 
par,  the  bond  must  rise  to  par;  if  bought  above  par,  it  must 
sink  to  par.  This  gradual  change  is  the  resultant  effect  of 
two  opposing  forces,  the  interest  earned  tending  to  increase 
the  investment  value,  while  the  payment  of  coupons  reduces 
the  investment  value.  At  any  intermediate  moment  between 
these  two  points  there  is  an  investment  value  which  can  be 
calculated,  and  which  is  just  as  true  as  the  original  cost  and 
the  par.  In  fact,  these  latter  are  merely  special  cases  of  in- 
vestment value ;  the  investment  value  at  the  date  of  purchase 
is  cost,  and  at  the  date  of  maturity  it  is  par. 

§  105.     Intermediate  Value  of  Bonds 

The  gradual  change  in  investment  value  of  bonds  be- 
tween purchase  and  maturity  is  ignored  by  some  investors, 
who,  during  the  whole  period,  use  either  the  original  cost  or 
the  par  value.  In  the  former  case  they  suppose  that  the  in- 
vestment value  remains  at  its  original  figure  until  the  very 
day  of  maturity,  and  is  then  instantly  changed  to  par,  either 
by  a  loss  of  all  of  the  premium  or  by  a  sudden  gain  of  all  of 
the  discount.  Those  who  use  par  as  the  investment  value 
also  assume  that  there  is  this  sudden  change  of  value,  the 
difference  being  that  the  change  occurred  at  the  instant  of 
purchase  instead  of  at  maturity.  These  methods  of  treat- 
ment are  manifestly  fictitious  and  unreal,  and  are  only  re- 
sorted to  on  account  of  the  labor  involved  in  computing  in- 
termediate values.  Experience  would  tell  us,  if  theory  did 
not,  that  there  is  no  such  violent  change.  The  cost  and  the 
par  value,  while  entirely  correct  at  the  beginning  and  at  the 
end,  respectively,  of  the  period  of  ownership,  are  entirely 
incorrect  during  the  interim. 


8o  THE  MATHEMATICS  OF  INVESTMENT 

§  1 06.     True  Investment  Basis  for  Bonds 

The  true  standard  of  investment  value  for  bonds  is  the 
present  worth,  at  compound  interest,  of  all  recipiends,  or 
sums  of  cash  to  be  received,  whether  such  sums  be  called 
coupons  or  principal.  Neither  the  original  cost  of  a  bond 
nor  its  ultimate  par  is  a  proper  permanent  investment  basis. 
The  bond  should  enter  into  the  accounts  at  cost,  which  is  a 
fact,  and  should  go  out  of  the  accounts  at  par,  which  is  an- 
other fact.  During  the  interim,  the  change  from  cost  to  par 
should  take  place  gradually  by  the  processes  of  amortization 
or  accumulation,  at  the  rate  of  the  true  interest  on  the 
original  investment. 

§  107.    Various  Bond  Values 

There  are  thus  three  values  in  the  life  of  a  bond  which 
resemble  three  tenses  in  grammar :  The  past  tense  represents 
the  cost,  that  is,  the  amount  originally  paid ;  the  future  tense 
represents  the  par,  which  is  the  amount  ultimately  to  be  re- 
ceived; while  the  present  tense  represents  the  investment 
value,  intermediate  between  the  values  of  the  past  and  future, 
except  in  the  special  case  of  a  bond  bought  at  par. 

There  is  also  a  fourth  value  of  a  bond,  that  is,  the 
amount  which  might  be  obtained  on  sale  at  the  present  time. 
This  is  the  market  value,  and  is  a  matter  of  judgment, 
opinion,  and  inference.  Although  the  market  value  of  a 
bond  has  great  utility  in  some  respects,  it  has  no  place, 
strictly  speaking,  in  accounts  kept  with  regard  to  invest- 
ments. It  is  not  an  act  or  a  fact  of  the  business;  it  is  a 
statement  of  what  might  be  done.  The  market  value  con- 
templates a  possibility,  or  a  probability — ^but  never  an 
actuality,  in  so  far  as  the  accounts  are  concerned,  unless  a 
sale  is  actually  consummated.  If  an  investor  has  had  an 
opportunity  to  make  a  sale  of  a  bond,  but  has  allowed  it  to 
pass  by,  the  mere  fact  that  he  has  been  offered  such  an  op- 


BONDS  AND  BOND  ACCOUNTS  8l 

portunity  to  sell  has  not  the  slightest  effect  on  his  financial 
status. 

§  1 08.    Investment  Value  the  True  Accounting  Basis 

Unless  accounts  with  respect  to  bonds  and  similar  securi- 
ties are  kept  on  the  investment-value  basis,  an  investor  is 
unable  to  tell  whether  a  contemplated  selling  price  will  result 
in  a  loss  or  a  gain.  If  the  books  are  kept  on*  the  basis  of 
par,  every  sale  above  par  will  appear  as  a  gain,  even  though 
it  may  be  a  losing  bargain;  while  a  comparison  with  the 
Driginal  cost  will  be  equally  delusive  and  unsatisfactory. 


CHAPTER  X 

VALUATION    OF   BONDS 

§  109.     Cash  Rate  and  Income  Rate  of  Bonds 

With  respect  to  all  bonds  bought  above  or  below  par, 
there  are  always  two  rates  of  interest  involved:  first,  a 
nominal  or  cash  rate,  which  is  a  certain  percentage  of  par, 
and  which  is  indicated  by  the  coupons ;  and  second,  an  effec- 
tive or  income  rate,  which  is  a  certain  percentage  of  the 
amount  originally  invested  and  remaining  invested.  For  the 
sake  of  greater  clearness,  we  shall  use  the  terms  cash  rate 
and  income  rate,  since  they  are  more  readily  understood 
than  the  terms  nominal  and  effective.  The  symbols  c  and 
i  will  respectively  designate  the  cash  rate  and  the  income 
rate.  1  +  i  is  the  ratio  of  increase  as  heretofore.  The 
symbol  1  +  c  will  not  be  required,  since  c  is  not  an  accumula- 
tive rate,  but  merely  an  annuity  purchased  with  the  bond, 
the  number  of  periods  of  the  annuity  being  the  same  as  the 
number  of  coupons  attached  to  the  bond.  The  difference  of 
rates  is  c  —  i,  or  i  —  c. 

§110.    Elements  of  a  Bond 

In  a  bond  purchased  above  or  below  par,  we  have,  there- 
fore, the  following  elements :  the  par,  or  principal,  payable 
after  n  periods ;  an  annuity  of  c  per  cent  of  par  for  n  periods ; 
and  a  ratio  of  increase,  1  +  i.  With  these  elements  given, 
there  are  two  distinct  methods  for  finding  the  value  of  the 
entire  security,  and  these  must  give  the  same  result. 

82 


VALUATION  OF  BONDS  83 

§111.     Valuation  of  Bonds — First  Method 

As  an  illustration  of  this  method,  let  us  take  the  case 
of  a  7%  bond,  having  25  years  (50  periods)  to  run,  interest 
payable  semi-annually,  the  par  being  $1,000.  Suppose  that 
it  is  required  to  compute  the  value  of  the  bond  at  the  be- 
ginning of  its  first  interest  period.  This  present  value  is 
composed  of  two  parts :  (a)  the  present  worth  of  $1,000 
due  50  periods  hence;  and  (b)  the  present  worth  of  an  an- 
nuity of  $35  for  50  terms.  We  cannot  ascertain  the  value 
of  these  two  parts  until  we  know  the  income  rate  current 
upon  securities  of  a  similar  grade.  Let  us  assume  that  this 
income  rate  is  3%  per  period,  or  what  is  usually  called  a 
6%  basis.    The  ratio  of  increase  is  thus  1.03  per  period. 

§  112.     (a)  Finding  Present  Worth  of  Principal 

The  first  part  of  the  solution  is  to  find  the  present  worth 
of  $1,000  due  in  50  periods,  at  3%  per  period.  In  §  52, 
we  have  found  the  present  worth  of  $1.00,  under  the  same 
conditions,  to  be  $.2281071 ;  hence  the  similar  present  worth 
of  $1,000  is  $228.1071.  This  result,  it  will  be  noticed,  has 
not  the  slightest  reference  to  the  7%  rate  of  the  bond.  For 
the  purposes  of  the  first  part  of  the  solution,  the  cash  or 
coupon  rate  is  absolutely  immaterial;  the  bond  might  be 
equally  well  a  10%  bond  or  a  0%  bond,  in  the  latter  case 
bearing  no  coupons  at  all. 

§113.     (b)  Present  Worth  of  Coupons 

We  next  have  to  find  the  present  value  of  an  annuity  of 
$35  for  50  terms  at  3%.  In  §  73,  we  found  the  present  value 
of  a  similar  annuity  of  $1.00  to  be  $25.72976  +.  An 
annuity  of  $35,  therefore,  has  a  present  value  of  $900.5417. 
Hence,  we  have  the  following : 


84 


THE  MATHEMATICS  OF  INVESTMENT 

Present  worth  of  the  par $228.1071 

Present  worth  of  the  coupons 900.5417 

Present  worth  of  the  entire  bond $1,128.6488 


The  ordinary  tables,  which  give  the  values  of  a  $100  bond 
only,  read  $112.86,  which  is  the  same  as  the  above,  rounded 
off.  The  above  computation  gives  a  result  which  is  correct 
to  the  nearest  cent  on  $100,000,  viz. :  $112,864.88. 


§114.     Schedule  of  Evaluation 

In  order  to  present  the  subject  still  more  clearly,  in  a 
schedule  form,  let  it  be  required  to  find  the  value,  as  at  Jan- 
uary 1,  1913,  of  a  7%  bond  for  $1,000,  interest  payable 
semi-annually,  due  at  January  1,  1915,  the  income  rate  being 
3%  per  period.  In  §  30,  we  have  found  that  the  present 
worth  of  $1.00.  for  1,  2,  3,  and  4  periods  is  $.970874, 
$.942596,  $.915142,  and  $.888487,  respectively.  The  re- 
spective present  worths  of  $35.00  are,  therefore,  $33.980590, 
$32.990860,  $32.029970,  and  $31.097045.  The  following 
schedule  may  then  be  formed : 


Periods  from 

Present 

Items  to  be 

Dates  of 

Jan.  1,  1013, 

Worth 

Evaluated 

Maturity- 

to  Dates  of 

at  Jan.  1, 

Maturity 

1913 

Coupon,  $35 
35 
35 
35 

July 
January 

July 

January 

1,  1913 
1,  1914 
1,  1914 
1,  1915 

1 

2 
3 
4 

$33.980590 
32.990860 
32.029970 
31.097045 

Total  . 

. .    $130.098465' 

Par,  $1,000 

January 
rotal 

1,  1915 

4 

888.487 

Grand  1 

.$1,018.585465 

VALUATION  OF  BONDS  85 

The  total  present  value  of  the  four  coupons 
($130.098465)  could  have  been  found  by  one  operation,  as 
was  done  in  the  preceding  section,  and  this  is  the  usual 
method  of  finding  the  present  worth  of  an  annuity.  The 
foregoing  schedule,  however,  sets  forth  the  details  clearly, 
although  it  is  not  a  practicable  method  of  evaluation  when 
the  number  of  coupons  is  large. 

§  115.    Valuation  of  Bonds — Second  Method 

In  illustration  of  this  method,  we  shall  assume  the  same 
facts  as  presented  in  §  111.  Each  semi-annual  payment  of 
$35  may  be  considered  as  made  up  of  two  parts :  $30  and 
$5.  The  $30  is  the  income  on  the  $1,000  par  value  at  the 
assumed  semi-annual  income  rate  of  3%.  We  may  disre- 
gard this,  and  consider  only  the  $5,  which  is  a  surplus  over 
and  above  the  income  rate,  and,  in  fact,  is  an  annuity  which 
must  be  paid  for  and  which  is  represented  by  the  premium 
paid  on  the  bond.  Having  devoted  $30  to  the  payment  of 
our  expected  income-rate  on  par,  we  have  a  remainder  of 
$5,  the  difference  in  rates  per  period ;  this  annuity  of  $5,  in 
excess  of  the  income  rate,  is  a  semi-annual  benefit  the  value 
of  which  is  to  be  ascertained. 

We  have  already  found  the  present  value  of  an  annuity 
of  $1.00  for  50  terms  at  3%  to  be  $25.72976.  The  present 
value  of  a  similar  annuity  of  $5.00  would  therefore  be 
$128.6488,  which  is  the  premium  and  which  agrees  with 
the  result  found  in  §  113.  The  method  is  not  only  quicker 
than  the  first  method  presented,  but  also  often  gives  one 
more  place  of  decimals. 

§116.    Evaluation  when  Cash  Rate  Is  Less  than  Income 
Rate 

In  the  case  of  a  bond  sold  below  par,  and  where,  ac- 
cordingfly,  the  cash  rate  is  less  than  the  income  rate,  the 


86 


THE  MATHEMATICS  OF  INVESTMENT 


same  procedure  is  followed  for  finding  the  present  worth 
of  an  annuity  of  the  difference  in  rates.  In  the  above  illus- 
tration, if  the  bond  had  a  cash  rate  of  5%  instead  oi  7%, 
the  annuity  to  be  evaluated  would  still  be  $5  (that  is,  $30 
less  $25).  In  this  case,  however,  the  value  of  the  annuity 
($128.6488)  would  have  to  be  subtracted  from  the  par, 
giving  $871.3512  as  the  value  of  a  5%  bond,  due  in  25 
years  and  having  an  income  rate  of  3%  per  period.  This 
would  be  commonly  known  as  a  6%  basis,  although  the 
effective  annual  income  is  6.09%,  as  pointed  out  in  §  91. 

§117.     Second  Method  by  Schedule 

As  a  further  illustration  of  the  second  method  of 
evaluation,  let  us  take  the  case  of  the  bond  described  in 
§  114.    Under  the  second  method  the  schedule  would  be : 


Differences 

Between  Cash  and 

Income  Rates 

(c-i) 

Dates  of 
Maturity 

Periods  from 

Jan.  1, 1913, 

to  Dates  of 

Maturity 

Present 
Worth 
at  Jan.  1, 
1913       • 

$5 
5 
5 
5 

July          1,  1913 
January   1,  1914 
July          1,  1914 
January   1,  1915 

al 

1 

2 
3 
4 

$4.854370 
4.712980 
4.575710 
4.442435 

Tot 

.  .$18.585495 

The  premium  above  found  disagrees  slightly  with  that 
shown  in  §  114,  since  in  the  latter  case  there  is  a  loss  of 
three  decimal  places  in  finding  the  present  worth  of  the 
$1,000  par  value.  In  examining  the  above  schedule,  it  must 
be  borne  in  mind  that  the  total  can  be  ascertained  by  a 
single  operation,  and  that  the  details  are  here  presented  only 
for  the  sake  of  additional  clearness. 


VALUATION  OF  BONDS  87 

§  118.    Rule  for  Second  Method  of  Evaluation 

.  Since  the  second  method  is  superior  to  the  first,  it  will 
hereafter  be  considered  as  the  standard;  and  we  give  ac- 
cordingly the  following  rule:  The  premium  (or  discount) 
on  a  bond  bought  above  (or  below)  par  is  the  present  worth, 
at  the  income  rate,  of  an  annuity  equal  to  the  difference  be- 
tween the  cash  and  income  rates  for  the  life  of  the  bond. 

§  119.     Principles  of  Investment 

We  have  found  the  value  of  a  7%  bond  for  $1,000, 
paying  6%  (semi-annually),  due  in  25  years,  to  be 
$1,128.65  to  the  nearest  cent.  This  is  the  amount  which 
must  be  invested  if  the  6%  income  is  to  be  secured.  At  the 
end  of  the  first  half-year,  the  holder  of  the  bond  receives,  as 
income,  3%  interest  on  the  $1,128.65  originally  invested, 
which  is  $33.86.  But  he  actually  collects  $35.00,  and  after 
deducting  $33.86  as  revenue,  there  remains  $1.14,  which 
must  be  applied  in  amortizing  the  premium.  This  will 
leave  the  value  of  the  bond  at  the  end  of  the  first  half-year, 
at  the  same  income  rate,  $1,127.51.  If  our  operations  have 
been  correct,  the  value  of  a  7%  bond  to  net  6%  (payable 
semi-annually),  having  24%  years  or  49  periods  to  run, 
will  be  $1,127.51.  To  test  this,  and  to  exemplify  the  method 
through  the  use  of  logarithms,  the  entire  operation  is  pre- 
sented in  the  following  section. 

§  120.     Solution  by  Logarithms 

The  logarithm  of  1  is zero 

The  logarithm  of  1.03  is .01283722 

The  logarithm  of  1.03*'  is  therefore _.6290238 

The  logarithm  of  (1  ^  1.03"")  is  therefore. .  1.3709762 

We  find  that  the  logarithm  of  .23495  is 1.3709754 

Remainder 8 

This  gives  the  additional  decimal  figures  02. 


88 


THE  MATHEMATICS  OF  INVESTMENT 


Hence,  $.2349502  is  the  present  value  of  $1.00  at  3% 
per  period  for  49  periods.  The  compound  discount  is  there- 
fore $.7650498,  and  this  divided  by  the  single  rate  of  in- 
terest, 3%,  gives  the  result  $25.50166,  which  is  the  present 
value  of  an  annuity  of  $1.00  per  period.  The  difference  be- 
tween the  cash  and  income  rates  is  $5,  i.e.,  $35  —  $30. 
Therefore,  the  present  value  of  a  $5  annuity  for  49  periods 
at  3%  would  be  $127,508,  or,  rounded  off,  $127.51,  which 
is  the  premium  desired.  Adding  this  to  the  par,  we  have 
$1,127.51,  which  agrees  with  the  result  obtained  in  §  119. 


§  121.     Amortization  Schedule 

When  bonds  are  purchased  for  investment  purposes,  a 
Schedule  of  Amortization  should  be  constructed,  showing 
the  gradual  extinction  of  the  premium  by  the  application  of 
the  surplus  interest.  The  form  shown  below  is  recom- 
mended for  this  purpose,  although  it  is  merely  suggestive 
and  not  complete.  The  calculations  should  be  continued  to 
the  date  of  maturity,  and  at  intervals  corrected  in  the  last 
figure  by  a  fresh  logarithmic  computation. 


Schedule  of  Amortization 

7%  Bond  of  the. , ,  payable  January  1, 

1939.     Net  6%.     J  J. 


Date 

Total 
Interest 

7% 

Net 
Income 

Amortiza- 
tion 

Book 
Value 

Par 

1914,  Jan.  1 

Cost 

..$1  128  65 

$1,000.00 

Julyl 

1915,  Jan.  1 

Julyl 

$35.00 
35.00 
35.00 

$33.86 
33.83 
33.79 

$1.14 
1.17 
1.21 

1,127.51 
1,126.34 
1,125.13 

Strictly  speaking,  the  net  income  rate  is  not  6%   per 
annum,  but  3%  for  each  semi-annual  period,  or  an  effective 


VALUATION  OF  BONDS 


89 


annual  rate  of  6.09%.  The  column  headed  "Total  Interest" 
could  be  changed  to  "Cash  Receipts,"  and  the  term  "Book 
Value"  might  also  be  called  "Investment  Value." 

§  122.    Use  of  Schedules  in  Accountancy 

The  foregoing  schedule  is  the  source  of  the  entry  which 
should  be  made  each  half-year  for  "writing  off"  the  premium 
or  "writing  up"  the  discount,  in  order  that  at  maturity  the 
bond  may  stand  exactly  at  par.  Two  other  schedules  are 
set  forth  below,  in  which  the  semi-annual  steps  in  the  chang- 
ing value  of  the  bond  are  shown  in  detail  from  the  date  of 
purchase  until  maturity,  one  schedule  being  for  a  bond 
bought  above  par,  and  the  other  for  a  bond  bought  below 
par.  Since  the  formation  of  schedules  is  the  basis  of  the 
accountancy  of  amortized  securities,  we  shall  present  the 
same  material  in  various  forms,  and  shall  attach  to  the 
schedules  the  letters  (A),  (B),  etc.,  for  the  purposes  of 
ready  reference. 

Schedule  (A) — ^Amortization 

5%  Bond  of  the ,. .,  payable  May  1, 

1919.    M  N. 


Date 

Total 

Interest 

5< 

Net 
Income 

Amortiza- 
tion 

Book 
Value 

Par 

1914,  May  1 
Nov.  1 

1915,  May    1 
Nov.  1 

1916,  May   1 
Nov.  1 

1917,  May   1 
Nov.  1 

1918,  May   1 
Nov.  1 

1919,  May   1 

Cost 

$104,491.29 
104,081.12 
103,662.74 
103,236.00 
102,800.72 
102,356.73 
101,903.86 
101,441.94 
100,970.78 
100,490.20 
100,000.00 

$100,000.00 

$  2,500.00 
2,500.00 
2,500.00 
2,500.00 
2,500.00 
2,500.00 
2,500.00 
2,500.00 
2,500.00 
2,500.00 

$  2,089.83 
2,081.62 
2,073.26 
2,064.72 
2,056.01 
2,047.13 
2,038.08 
2,028.84 
2,019.42 
2,009.80 

$    410.17 
418.38 
426.74 
435.28 
443.99 
452.87 
461.92 
471.16 
480.58 
490.20 

$25,000.00 

$20,508.71 

$4,491.29 

90 


THE  MATHEMATICS  OF  INVESTMENT 


Schedule  (B) — Accumulation 

3%  Bond  of  the ,  payable  May  1, 

1919.     M  N. 


Date 

Total 
Interest 

3^ 

Net 

Income 

4^ 

Accumula- 
tion 

Book 
Value 

Par 

1914,  May    1 

$95,508.71 

$100,000.00 

Nov.    1 

$  1,500.00 

$  1,910.17 

$    410.17 

95,918.88 

1915,  May    1 

1,500.00 

1,918.38 

418.38 

96,337.26 

Nov.    1 

1,500.00 

1,926.74 

426.74 

96,764.00 

1916,  May    1 

1,500.00 

1,935.28 

435.28 

97,199.28 

Nov.    1 

1,500.00 

1,943.99 

443.99 

97,643.27 

1917,  May    1 

1,500.00 

1,952.87 

452.87 

98,096.14 

Nov.    1 

1,500.00 

1,961.92 

461.92 

98,558.06 

1918,  May    1 

1,500.00' 

1,971.16 

471.16 

99,029.22 

Nov.    1 

1,500.00 

1,980.58 

480.58 

99,509.80 

1919,  May    1 

1,500.00 

1,990.20 

490.20 

100,000.00 

$15,000.00 

$19,491.29 

$4,491.29 

§  123.     Book  Values  in  Schedules 

In  the  foregoing  two  schedules,  (A)  and  (B),  it  will 
be  observed  that  at  any  given  date  the  book  value  in 
Schedule  (A)  is  always  exactly  as  much  above  par  as  the 
book  value  in  Schedule  (B)  is  below  par.  During  any 
given  period,  the  "amortization"  and  the  "accumulation" 
are  exactly  the  same  in  both,  being  deducted  in  Schedule 
(A)  and  added  in  Schedule  (B). 


§  124.     Checks  on  Accuracy  of  Schedules 

There  are  three  internal  checks  which  are  of  value  in 
verifying  the  accuracy  of  the  schedules.  For  example,  in 
Schedule  (B),  the  following  facts  may  be  observed: 

(1)  The  total  interest  plus  the  total  accumulation  equals 
the  total  net  income. 

(2)  The  total  accumulation  equals  the  par  less  the  book 
value;  or,  in  other  words,  it  equals  the  inaugural  discount. 


VALUATION  OF  BONDS 


91 


(3)  Each  item  of  accumulation  equals  the  preceding  one 
multiplied  by  the  semi-annual  ratio  of  increase  1.02,  the 
semi-annual  net  income  being  2%.    That  is : 

$461.92X1.02  =  $471.16 

$471.16  X  1.02  =  $480.68 

etc. 

In  some  instances  in  these  computations,  there  will  be  an 
apparent  error  of  one  cent,  which  is  accounted  for  by  the 
fact  that  the  number  of  decimal  places  is  not  carried  out 
sufficiently  far. 

§  125.    Tables  Derivable  from  Bond  Values 

The  figures  in  the  column  headed  "Book  Value"  might 
be  taken  from  tables  of  bond  values  published  in  book  form. 
If  Sprague's  Eight-Place  Bond  Tables  were  used,  and  if 
the  column  "Book  Value"  were  copied  directly  from  the 
tables,  the  other  columns  could  be  derived  by  the  processes 
of  addition  or  subtraction.  The  result  arrived  at  by  this 
method  would  be  exactly  the  same  as  the  results  shown  in 
Schedules  (A)  and  (B).  The  successive  amounts  of 
amortization  or  accumulation  would  be  found  by  finding 
the  differences  between  successive  book  values;  while  the 
net  income  for  any  period  would  be  found  by  either  adding 
the  accumulation  to  the  total  interest,  or  by  deducting  the 
amortization  from  the  total  interest. 

§  126.     Methods  of  Handling  Interest 

It  will  be  observed  that  in  Schedules  (A)  and  (B),  the 
entire  interest  is  accounted  for,  both  in  the  case  of  the 
interest  on  par  plus  premium,  and  also  in  the  case  of  the  in- 
terest on  par  minus  discount.  We  may  easily  construct  the 
schedules  so  as  to  eliminate  the  par  and  the  interest  thereon 


92 


THE  MATHEMATICS  OF  INVESTMENT 


at  the  rate  i.  In  this  manner  we  would  have  to  deal  only 
with  the  surplus  interest  or  the  deficient  interest,  according 
to  the  theory  explained  in  §  115.  Since  this  method  may  be 
preferable  for  some  forms  of  accounts,  a  new  schedule  is 
presented  below,  based  on  the  same  facts  as  those  shown  in 
Schedule  (A)  : 


Schedule  (C) — Amortization;  Premium  Only 


Date 

Surplus 
Interest 
on  Par 

1% 

Interest 

on 
Premium 

4^ 

Amortiza- 
tion 

Premium 

1914,  May 

$4,491.29 

Nov. 

$   500.00 

$  89.83 

$    410.17 

4,081.12 

1915,  May 

500.00 

81.62 

418.38 

3,662.74 

Nov. 

500.00 

73.26 

426.74 

3,236.00 

1916,  May 

500.00 

64.72 

435.28 

2,800.72 

Nov. 

500.00 

56.01 

443.99 

2,356.73 

1917,  May 

500.00 

47.13 

452.87 

1,903.86 

Nov. 

500.00 

38.08 

461.92 

1,441.94 

1918,  May 

500.00 

28.84 

471.16 

970.78 

Nov. 

500.00 

19.42 

480.58 

490.20 

1919,  May 

500.00 

9.80 

490.20 

0. 

$5,000.00 

$508.71 

$4,491.29 

§  127.     Schedule  of  Bond  Values 

Another  way  of  setting  forth  the  value  of  bonds  at  the 
successive  interest  dates  is  shown  in  the  following  table, 
which  indicates  clearly  the  steps  taken  in  computing  the 
value.  This  table,  however,  is  not  nearly  so  compact  as  the 
preceding  ones,  and  for  this  reason  is  not  recommended,  for 
most  purposes.  We  will  take  as  an  illustration  Schedule 
(A),  shown  in  §122. 


VALUATION  OF  BONDS  93 

Value  of  bond  at  May  1,  1914  (cost) $104,491.29 

Amortization  for  ensuing  6  months : 
Nominal  interest  at  2%%  on 

$100,000.00 $2,500.00 

Effective  interest  at  2%  on 

$104,491.29 '. 2,089.83 

Difference,  being  the  amor- 
tization to  be  subtracted 
from  the  investment  value 410.17 

Value  of  bond  at  November  1,  1914 $104,081.12 

Amortization  for  ensuing  6  months : 
Nominal  interest  at  2^/2%  on 

$100,000.00 $2,500.00 

Effective  interest  at  2%  on 

$104,081.12   2,081.62 

Difference,  being  the  amor- 
tization to  be  subtracted 
from  the  investment  value , 418.38 

Value  of  bond  at  May  1,1915 $103,662.Y4 

etc.,  etc. 
A  slight  variation  of  the  above  form  is  to  put  all  of  the 
figures  of  the  schedule  in  one  column,  as  follows : 

Value,  May  1,  1914 $104,491.29 

Plus  effective  interest 2,089.83 

$106,581.12 
Minus  amortization 2,500.00 

Value,  November  1,  1914 $104,081.12 

Plus  effective  interest 2,081.62 

$106,162.74 

Minus  amortization 2,500.00 

Value,  May  1,  1915 $103,662.74 

etc.,  etc. 


94 


THE  MATHEMATICS  OF  INVESTMENT 


By  using  red  ink  for  the  subtrahends  (which  are  indi- 
cated by  italic  figures),  the  addition  and  subtraction  can  be 
performed  at  one  operation,  viz. : 

$104,491.29 

2,089.83 
2,500.00 

$104,081.12 
2,081.62 
2,500.00 

$103,662.74 
2,073.26 
2,500.00 

$103,236.00 
etc.,  etc. 

It  will  be  noticed  that  the  computation  of  the  interest 
may  be  done  without  using  any  other  paper.  Even  with  a 
fractional  rate,  such  as  2.7%  per  annum,  or  1.35%  per 
period,  the  1%,  the  .3%,  and  the  .05%  may  be  successively 
written  down  direct  without  further  computation.  For 
example : 

Assumed  inaugural  value $120,039.00 

1,200.39 
360.117 
60.019 
2,500.00 

Value  at  end  of  6  months ,. .  .$119,159,526 

1,191.595 
357.479 
59.580 
2,500.00 


Value  at  end  of  1  year $118,268,180 

etc.,  etc. 


CHAPTER  XI 

VALUATION  OF  BONDS   (Concluded) 

§  128.     Bond  Purchases  at  Intermediate  Dates 

It  has  hitherto  been  assumed  that  the  purchase  of  the 
bond  took  place  exactly  upon  an  interest  date.  In  the  vast 
majority  of  purchases,  however,  the  purchase  date  differs 
from  the  interest  date,  and  we  will  now  consider  cases  of 
this  character.  Let  us  suppose  that  the  interest  dates  are 
May  1  and  November  1,  whereas  the  purchase  took  place  on 
July  1,  after  one-third  of  the  interest  period  had  elapsed. 
The  business  custom  is  to  adjust  the  matter  as  follows :  The 
buyer  pays  to  the  seller  the  (simple)  interest  accrued  for  the 
two  months,  acquiring  thereby  the  full  interest  rights,  which 
will  fall  due  on  November  1,  and  the  premium  (or  the  dis- 
count, as  the  case  may  be)  is  also  considered  as  vanishing 
by  an  equal  portion  each  month,  so  that  one-third  of  the  half- 
yearly  amortization  takes  place  by  July  1.  Taking  as  an 
illustration  the  bond  considered  in  Schedule  (A)  (§122), 
the  amortization  from  May  1,  1914,  to  November  1,  1914, 
is  $410.17;  the  amortization  up  to  July  1  would  therefore 
be  one-third  of  this  amount,  or  $186.72.  The  book  value  at 
July  1  is  $104,491.29  minus  $136.72,  plus  $833.33  (the 
accrued  interest  for  two  months),  giving  a  net  figure  of 
$105,187.90.  This  last  amount  is  called  the  nat  price,  that 
is,  it  is  the  price  including  interest ;  if  the  interest  is  not  in- 
cluded, the  price  is  said  to  be  at  so  many  per  cent  and 
interest.  These  are  the  two  methods  in  most  common  use 
for  indicating  the  prices  of  bonds.    The  flat  price  as  above 

95 


C|6  THE  MATHEMATICS  OF  INVESTMENT 

computed  might  also  have  been  obtained  in  the  following 
manner : 

To  the  value  on  May  1,  1914 $104,491.29 

add  simple  interest  thereon  for  2  months  at  4%, 

which  is  the  effective  income  rate 696.61 

giving  the  flat  price  at  July  1,  1914 $105,187.90 


§  129.     Errors  in  Adjusting  Bond  Prices 

This  practice  of  adjusting  the  price  of  bonds  at  inter- 
mediate dates  by  simple  interest  is  conventionally  correct, 
but  is  scientifically  inaccurate,  and  always  works  a  slight 
injustice  to  the  buyer.  The  seller  is  having  his  interest 
compounded  at  the  end  of  two  months  instead  of  six  months, 
and  receives  a  benefit  therefrom  at  the  expense  of  the  buyer. 
It  will  be  readily  seen  that  the  buyer  does  not  net  the  effec- 
tive rate  of  4%  semi-annually  on  his  investment  of  $105,- 
187.90.  In  order  to  give  both  buyer  and  seller  a  return  at 
the  effective  rate  of  2%  semi-annually  (or  4.04%  annually), 
with  a  bimonthly  conversion  for  the  seller  and  a  four-month- 
ly conversion  for  the  buyer,  the  true  price  would  be  $105,- 
183.31.*  In  practice,  however,  for  any  time  under  six 
months,  simple  interest  is  generally  used,  to  the  slight  dis- 
advantage of  the  buyer,  who  may  claim  that  the  value  at 
November  1,  ($104,081.12)  +  interest  due  ($2,500.00), 
should  have  been  discounted  at  4%.  This  would  give  $106,- 
581.12 -^- 1.01  >^,  or  $105,178.74.  This  latter  figure  is  al- 
most exactly  as  much  too  low  ($4.57)  as  the  $105,187.90 
is  too  high  ($4.59). 


•This  price  is  found  by  finding  the  cube  root  of  1.02,  which  is  1.00662271. 
This  last  figure  is  the  rate  for  a  two-months  period  at  the  effective  rate  of  2% 
semi-annually,  or  4.04%  annually.  When  the  value  at  May  1  ($104,491.29)  is 
multiplied  by  this  figure,  the  result  is  $105,183.31,  which  is  the  true  price  on  an 
effective  income  basis  of  4.04%  annually. 


VALUATION  OF  BONDS 


97 


§  130*     Schedule  of  Periodic  Evaluation 

The   schedule   would    therefore,   in  practice,   read   as 
follows : 


Schedule  (D) — Periodic  Valuation;  Simple  Interest 


Date 

Total 
Interest 

5^ 

Net 
Income 

A% 

Amortiza- 
tion 

Book 
Value 

Par 

1914,  July    1 
Nov.  1 

Cost 

$104,354.57 
104,081.12 

$100,000.00 

$  1,666.67 

$  1,393.22 

$  273.45 

1915,  May   1 

2,500.00 

2,081.62 

418.38 

103,662.74 

Nov.  1 

2,500.00 

2,073.26 

426.74 

103,236.00 

1916,  May   1 

2,500.00 

2,064.72 

435.28 

102,800.72 

Nov.  1 

2,500.00 

2,056.01 

443.99 

102,356.73 

1917,  May   1 

2,500.00 

2,047.13 

452.87 

101,903.86 

Nov.  1 

2,500.00 

2,038.08 

461.92 

101,441.94 

1918,  May   1 

2,500.00 

2,028.84 

471.16 

100,970.78 

Nov.  1 

2,500.00 

2,019.42 

480.58 

100,490.20 

1919,  May   1 

2,500.00 

2,009.80 

490.20 

100,000.00 

$24,166.67 

$19,812.10 

$4,354.57 

§  131.    Objection  to  Valuation  on  Interest  Dates 

The  interest  dates  may  not  always  be  the  most  con- 
venient dates  for  periodical  valuation.  In  the  case  of  an 
investment  consisting  of  several  kinds  of  bonds,  there  would 
generally  be  some  interest  coupons  falling  due  in  every 
month  of  the  year,  and  yet  on  a  certain  annual  or  semi- 
annual date  the  entire  holdings  must  be  simultaneously 
valued,  irrespective  of  the  varying  interest  dates.  In  cases 
of  this  kind,  it  will  therefore  be  convenient  if  the  schedules 
can  be  arranged  in  such  a  manner  that,  without  recalculation, 
every  book  value  will  be  ready  to  place  in  the  balance  sheet. 
Fortunately,  this  is  easier  than  would  be  supposed. 


98  THE  MATHEMATICS  OF  INVESTMENT 

§  132.     Interpolation  Method  of  Periodic  Valuation 

As  an  illustration,  we  will  again  take  the  bond  described 
in  §  130 ;  but  we  will  now  assume  that  the  investor  closes 
his  books  on  the  last  days  of  June  and  December.  We  will 
suppose  that  the  purchase  is  made  on  August  1, 1914.  Since 
August  1  is  midway  between  May  1  and  November  1,  the 
price  must  be  adjusted  as  shown  in  §  128.  The  price  at 
August  1  would  therefore  be  midway  between  $104,491.29 
and  $104,081.12 — namely,  $104,286.20 — plus,  of  course, 
the  accrued  interest  ($1,250.00),  this  being  the  customary, 
not  the  theoretical,  method.  The  value  at  November  1  need 
not  enter  into  the  schedule,  but  we  must  compute  the  De- 
cember 31  value  in  the  same  manner  as  we  found  the  July  1 
value  in  §  128.  One-third  of  the  difference  between  $104,- 
081.12  and  $103,662.74,  or  $418.38,  is  $139.46;  $104,- 
081.12  — $139.46  =  $103,941.66.  Our  schedule  so  far, 
the  headings  being  the  same  as  in  §  130,  reads : 

1914,  Aug.    1  Cost $104,286.20    $100,000.00 

Dec.  31    $2,083.33    $1,738.79    $344.54      103.941.66 

Proceeding  in  the  same  way  to  find  the  value  on  June  30, 

1915,  from  those  on  May  1  and  November  1,  we  get  $103,- 
520.49.  To  these  values  at  dates  when  interest  does  not 
fall  due,  there  must  be  added  the  accrued  interest  to  find  the 
total  values.  This  method  of  finding  the  value  of  bonds  be- 
tween interest  dates  is  called  interpolation. 

§  133.     Multiplication  Method  of  Valuation 

There  is  another  method  of  finding  the  intermediate 
values,  however,  which  might  be  called  the  multiplication 
method.  Having  found  the  value  at  December  31  to  be 
$103,941.66,  the  interest  for  six  months  thereon  at  4%  is 
$2,078.83,    which,    subtracted    from    the   coupon    interest 


VALUATION  OF  BONDS  99 

($2,500.00),  gives  as  the  amortization  $421.17.  This  latter 
amount,  written  off  from  $103,941.66,  gives  $103,520.49  as 
the  value  at  June  30,  which  is  precisely  the  same  result  as 
was  obtained  by  interpolation  between  May  1  and  November 
1  in  §  132.  In  practice,  the  method  of  multiplication  will 
be  found  more  convenient  than  the  method  of  interpolation. 
Having  once  adjusted  the  value  at  one  of  the  balancing 
periods,  we  can  derive  all  of  the  values  at  the  remaining 
balancing  periods  by  finding  the  net  income,  subtracting  it 
from  the  cash  interest  and  reducing  the  premium  by  the 
difference,  completely  ignoring  the  values  on  interest  days 
(M  N). 

§  134.     Computation  of  Net  Income  for  Partial  Period 

No  difficulty  arises  until  we  reach  the  broken  period, 
January  1  to  May  1,  1919.  Here  the  computation  of  the 
net  income  is  peculiar;  the  par  and  the  premium  must  be 
treated  separately.  The  net  income  on  $100,000.00  is  taken 
at  Yz  of  2%  for  the  ^  time,  giving  $1,333.33.  The 
premium.  $326.80,  however,  must  always  be  multiplied  by 
the  full  2%,  giving  $6.54.  Adding  $1,333.33  and  $6.54, 
we  have  $1,339.87,  which,  used  as  heretofore,  reduces  the 
principal  to  par.  The  reason  for  this  peculiarity  is  that 
$490.20  (the  premium  at  November  1,  1918),  and  not 
$326.80  (the  premium  at  December  31,  1918),  is  the  con- 
ventional premium  on  which  4%  is  to  be  computed.  Hence, 
instead  of  taking  $490.20  for  ^  of  a  period,  we  take 
$326.80  itself  for  a  whole  period;  these  two  methods  reach 
the  same  result,  since  $490.20  is  3/2  of  $326.80,  and  tv^^o- 
thirds  of  three-halves  is  unity.  In  other  words,  Yz  of 
a  dollar  for  a  whole  period  is  equivalent  in  value  to  the  whole 
of  the  dollar  for  ^  of  a  period.  On  the  basis  outlined,  the 
completed  schedule  would  therefore  be  as  follows : 


lOO  THE  MATHEMATICS  OF  INVESTMENT 

Schedule  (E) — Periodic  Valuation  by 
Multiplication 


Date 

Total 
Interest 

Net 

Income 

4% 

Amortiza- 
tion 

Book 
Value 

Par 

1914,  Aug.     1 

Cost 

$104,286.20 
103,941.66 
103,520.49 
103,090.90 
102,652.72 
102,205.77 
101,749.89 
101,284.89 
100,810.59 
100,326.80 
100,000.00 

$100,000.00 

Dec.  31 

1915,  Jun.  30 
Dec.  31 

1916,  Jun.  30 
Dec.  31 

1917,  Jun.  30 
Dec.  31 

1918,  Jun.  30 
Dec.  31 

1919,  May     1 

$  2,083.33 
2,500.00 
2,500.00 
2,500.00 
2,500.00 
2,500.00 
2,500.00 
2,500.00 
2,500.00 
1,666.67 

$23,750.00 

$  1,738.79 
2,078.83 
2,070.41 
2,061.82 
2,053.05 
2,044.12 
2,035.00 
2,025.70 
2,016.21 
1,339.87 

$   344.54 
421.17 
429.59 
438.18 
446.95 
455.88 
465.00 
474.30 
483.79 
326.80 

$19,463.80 

$4,286.20 

§  135.     Purchase  Agreements 

In  all  the  foregoing  examples  it  has  been  assumed  that 
the  bond  has  been  bought  "on  a  basis,"  which  means  that 
the  buyer  and  seller  have  agreed  upon  the  income  rate  which 
the  bonds  shall  pay,  and  that  from  this  the  price  has  been 
adjusted.  But  in  probably  the  majority  of  cases  the  bargain 
is  made  "at  a  price,"  and  then  the  income  rate  must  be  found. 
This  is  a  more  difficult  problem. 

§  136.     Approximation  Method  of  Finding  Income  Rate* 

The  best  method  of  ascertaining  the  basis,  when  the 
price  is  given,  is  by  trial  and  approximation — in  fact,  all 
methods  more  or  less  depend  upon  that.  The  ordinary 
tables  will  locate  several  figures  of  the  rate,  and  one  more 
figure  can  safely  be  added  by  simple  proportion.  But  it  is 
an  important  question  to  what  degree  of  fineness  we  should 
try  to  attain.     It  seems  to  be  the  consensus  of  opinion  and 

*For  a  new  method  of  approximation,  see  Chapter  XXIII. 


VALUATION  OF  BONDS  lOl 

practice  that  to  carry  the  decimals  to  hundredths  of  one  per 
cent  is  far  enough,  although  in  some  cases,  by  introducing 
eighths  and  sixteenths,  two-hundredths  and  four-hundredths 
may  be  required.  Sprague's  Tables,  by  the  use  of  auxiliary 
figures,  give  values  for  each  one-hundredth  of  one  per  cent. 

§  137.    Application  of  Method 

Let  us  suppose  that  $100,000  of  5%  bonds,  5  years  to 
run,  M  N,  are  offered  at  the  round  price  of  104%  on  May 
1,  1914.  It  is  evident  that  this  is  nearly,  but  not  quite,  a 
4%  basis.  Trying  a  3.99%  basis  we  find  that  the  premium 
is  $4,537.39,  which  is  further  from  the  price  than  is  $4,- 
491.29,  the  4%  basis.  Hence,  4%  is  the  nearest  basis  within 
1/100  of  one  per  cent.  In  fact,  by  repeated  trials,  we  find 
that  the  rate  is  about  .0399812  per  annum.  It  is  manifest 
that  such  a  ratio  of  increase  as  1.0199906  would  be  very  un- 
wieldy and  impracticable,  and  that  such  laborious  exactness 
would  be  intolerable.  Yet  here  we  have  paid  $104,500,  and 
the  nearest  admissible  basis  gives  $104,491.29;  what  shall 
be  done  with  the  odd  $8.71  ?  It  must  disappear  before  ma- 
turity, and  on  a  4%  basis  it  will  be  even  larger  at  maturity 
than  now.  Three  ways  of  ridding  ourselves  of  it  may  be 
suggested. 

§  138.     First  Method  of  Eliminating  Residues 

Add  the  residue  $8.71  to  the  first  amortization,  thereby 
reducing  the  value  to  an  exact  4%  basis  at  once.  In  Schedule 
(A) — shown  in  §  122 — the  first  amortization  would  be 
$418.88,  instead  of  $410.17.  This  is  at  the  income  rate  of 
about  3.983%  for  the  first  half-year  and  thereafter  at  4%. 
For  short  bonds  the  result  is  fairly  satisfactory. 

§  139.     Second  Method  of  Eliminating  Residues 

Divide  $8.71  into  as  many  parts  as  there  are  periods. 


I02 


THE  MATHEMATICS  OF  INVESTMENT 


This  would  give  $.87  for  each  period,  except  the  first,  which 
would  be  e$.88  on  account  of  the  odd  cent.  Set  down  the  4% 
amortization  in  one  column,  the  $.87  in  the  next,  and  the 
adjusted  figures  in  the  third : 


$410.17 

$.88 

$411.05 

418.38 

.87 

419.25 

426.74 

.87 

427.61 

435.28 

.87 

436.15 

443.99 

.87 

444.86 

452.87 

.87 

453.74 

461.92 

.87 

462.79 

471.16 

.87 

472.03 

480.58 

.87 

481.45 

490.20 

.87 

491.07 

The  following  will  then  be  the  schedule : 

Schedule  (F) — Elimination  of  Residues; 
Second  Method 


Date 

Total 
Interest 

Net 
Income 
4%i-) 

Amortiza- 
tion 

Book 
Value 

Par 

1914,  May   1 

$104,500.00 

$100,000.00 

Nov.  1 

$  2,500.00 

$  2,088.95 

$    411.05 

104,088.95 

1915,  May   1 

2,500.00 

2,080.75 

419.25 

103,669.70 

Nov.  1 

2,500.00 

2,072.39 

427.61 

103,242.09 

1916,  May    1 

2,500.00 

2,063.85 

436.15 

102,805.94 

Nov.  1 

2,500.00 

2,055.14 

444.86 

102,361.08 

1917,  May   1 

2,500.00 

2,046.26 

453.74 

101,907.34 

Nov.  1 

2,500.00 

2,037.21 

462.79 

101,444.55 

1918,  May   1 

2,500.00 

2,027.97 

472.03 

100,972.52 

Nov.  1 

2,500.00 

2,018.55 

481.45 

100,491.07 

1919,  May   1 

2,500.00 

2,008.93 

491.07 

100,000.00 

$25,000.00 

$20,500.00 

$4,500.00 

VALUATION  OF  BONDS  103 

In  this  schedule  the  income  rate  varies  from  3.997995^ 
to  3.99822%;  hence  the  approximation  is  sufficiently  close 
for  any  holdings,  except  large  ones  for  long  maturities. 

§  140.     Third  Method  of  Eliminating  Residues 

For  still  greater  accuracy,  we  may  divide  the  $8.71  in 
parts  proportionate  to  the  amortization.  The  amortization 
on  the  4%  basis  amounts  to  $4,491.29,  and  v^e  have  an 
extra  amount  of  $8.71  to  exhaust.  Dividing  the  latter  by 
the  former,  we  have  as  the  quotient  .00194,  which  is  the 
portion  to  be  added  to  each  dollar  of  amortization.  With 
this  we  form  a  table  for  the  9  digits : 

100194 
200388 
300582 
400776 
500970 
601164 
701358 
801552 
901746 

From  this  table  it  is  easy  to  adjust  each  item  of  amortiza- 
tion, writing  down,  for  example,  to  the  nearest  mill : 

410.17  418.38  426.74  435.28 


400.776 

400.776 

400.776 

400.776 

10.019 

10.019 

20.039 

30.058 

.100 

8.016 

6.012 

5.010 

.070 

.301 
.080 

.701 
.040 

.200 

410.97 

.080 

419.19  427.57  436.12 


I04 


THE  MATHEMATICS  OF  INVESTMENT 


The  respective  amounts  of  amortization,  in  Schedule  (G), 
vary  (at  the  most)  but  8  cents  from  those  shown  in  Schedule 
(F). 


Schedule  (G) — Elimination  of  Residues 
Third  Method 


Date 

Total 
Interest 

5^ 

Net 
Income 
4%(-) 

Amortiza- 
tion 

Book 
Value 

Par 

1914,  May   1 

$104,500.00 

$100,000.00 

Nov.   1 

$  2,500.00 

$  2,089.03 

$    410.97 

104,089.03 

1915,  May   1 

2,500.00 

2,080.81 

419.19 

103,669.84 

Nov.  1 

2,500.00 

2,072.43 

427.57 

103,242.27 

1916,  May   1 

2,500.00 

2,063.88 

436.12 

102,806.15 

Nov.  1 

2,500.00 

2,055.15 

444.85 

102,361.30 

1917,  May   1 

2,500.00 

2,046.25 

453.75 

101,907.55 

Nov.  1 

2,500.00 

2,037.18 

462.82 

101,444.73 

1918,  May   1 

2,500.00 

2,027.93 

472.07 

100,972.66 

Nov.  1 

2,500.00 

2,018.49 

481.51 

100,491.15 

1919,  May  1 

2,500.00 

2,008.85 

491.15 

100,000.00 

$25,000.00 

$20,500.00 

$4,500.00 

§  141.    Short  Terminals 

It  sometimes  happens  (though  infrequently)  that  the 
principal  of  a  bond  is  payable,  not  at  an  interest  date,  but 
from  one  to  five  months  later,  making  a  short  terminal 
period.  The  following  is  a  very  simple  method  of  obtain- 
ing the  present  value  in  this  case.  It  will  not  be  necessary 
to  demonstrate  it,  but  an  example  will  test  it. 

Suppose  the  5%  bond,  M  N,  yielding  4%,  bought  May 
1,  1914,  were  payable  October  1,  instead  of  May  1,  1919, 
that  is,  in  10  5/6  periods.     The  short  period  is  5/6.     The 


VALUATION  OF  BONDS  105 

short  ratio  (at  4%)  will  be  1.0166 J^.     The  short  interest 
(at  5%)  will  be  .02083 J^. 

We  first  ascertain  the  value  for  the  ten  full 

periods,  viz.,  for  $1 1.0449129* 

Add  to  this  the  short  interest 0208333 

1.0657462 
and  divide  by  the  short  ratio 1.0166667 

To  perform  this  division  it  will  be  easier  to  divide  3  times 
the  dividend  by  3  times  the  divisor. 

3.05  )  3.1972386  (  Quotient  1.0482750 
3.05 

1472 
1220 

2523 
2440 

838 
610 


2286 
2135 

151 

152 


Multiplying  down  by  the  usual  procedure,  we  have  the 
following  schedule: 

•See  Schedule  (A),   §  122. 


k 


I06  THE  MATHEMATICS  OF  INVESTMENT 

Schedule  (H) — Short  Terminals 


Date 

Total 
Interest 

5!< 

Net 

Income 

4^ 

Amortiza- 
tion 

Book 
Value 

Par 

1914,  May  1 

$104,827.50 

$100,000.00 

Nov.  1 

$  2,500.00 

$  2,096.55 

$  403.45 

104,424.05 

1915,  May   1 

2,500.00 

2,088.48 

411.52 

104,012.53 

Nov.  1 

2,500.00 

2,080.25 

419.75 

103,592.78 

1916,  May   1 

2,500.00 

2,071.86 

428.14 

103,164.64 

Nov.  1 

2,500.00 

2,063.29 

436.71 

102,727.93 

1917,  May    1 

2,500.00 

2,054.56 

445.44 

102,282.49 

Nov.  1 

2,500.00 

2,045.65 

454.35 

101,828.14 

1918,  May   1 

2,500.00 

2,036.56 

463.44 

101,364.70 

Nov.  1 

2,500.00 

2,027.29 

472.71 

100,891.99 

1919,  May   1 

2,500.00 

2,017.84 

482.16 

100,409.83 

Oct.    1 

2,083.33 

1,673.50 
$22,255.83 

409.83 

100,000.00 

$27,083.33 

$4,827.50 

§  142.     Rule  for  Short  Terminals 

Ascertain  the  value  of  the  bond  for  the  full  number  of 
periods,  disregarding  the  terminal.  To  this  value  add  the 
short  interest,  and  divide  by  the  short  ratio. 

It  may  be  remarked  that  this  same  process  applies  to 
short  initial  periods.  It  even  applies  to  bonds  originally 
issued  between  interest  dates,  and  also  maturing  between 
interest  dates ;  in  the  case  of  bonds  of  this  description,  the 
process  would  be  applied  twice. 

§  143.     Discounting 

Hitherto  we  have  calculated  the  value  of  the  bond  at  its 
earliest  date,  and  then  obtained  the  successive  values  at 
later  dates  by  multiplication  and  subtraction.  We  can  also 
work  backwards,  however,  obtaining  each  value  from  the 


VALUATION  OF  BONDS 


107 


next  later  one  by  addition  and  division.     Let  us  take,  for 
illustration,  the  bond  shown  in  Schedule  (A),  §  122: 

Principal  to  be  received  at  maturity,  May  1, 

1919   $100,000.00 

Coupon  to  be  received  at  May  1,  1919 2,500.00 

Total  amount  receivable  at  May  1,  1919 $102,500.00 

Discounted  value  at  November  1,  1918,  exclud- 
ing the  coupon  receivable  at  that  date,  found 
by  dividing  $102,500.00  by  1.02. $100,490.20 

Coupon  to  be  received  at  November  1,  1918 ....        2,500.00 

Total  value  at  November  1,  1918,  including  the 

coupon  receivable  at  that  date $102,990.20 

Discounted  value  at  May  1,  1918,  excluding  the 
coupon  receivable  at  that  date,  found  by  divid- 
ing $102,990.20  by  1.02 $100,9Y0.Y8 

etc.,  etc. 

In  this  manner  successive  terms  may  be  obtained  as  far  as 
desired. 

§  144.     Last  Half- Year  of  Bond 

In  the  last  half-year  of  a  bond,  its  value  should  be  dis- 
counted, and  not  found  as  in  §  128.  Thus,  if  the  bond  men- 
tioned in  §  128  were  sold  three  months  prior  to  maturity,  its 
value  would  be  found  by  dividing  $102,500.00  by  1.01, 
which  would  give  $101,485.15  "flat,"  equivalent  to  $100,- 
235.15  and  interest;  whereas  by  the  ordinary  rule  it  would 
be  $100,245.10  (that  is,  midway  between  $100,490.20  and 
$100,000.00).  The  theoretically  exact  value  (recognizing 
effective  rates,  which  is  never  done  in  business)  would  be 
$100,240.13.  This  is  found  by  multiplying  the  value  six 
months  prior  to  maturity  ($100,490.20)  by  the  square  root 
of  1.02,  this  root  to  ten  decimal  places  being  1.0099504938. 


Io8  THE  MATHEMATICS  OF  INVESTMENT 

The  product  is  $101,490.13,  which,  less  the  accrued  interest 
amounting  to  $1,250.00,  gives  $100,240.13.  To  "split  the 
difference"  would  be  an  easy  way  of  adjusting  the  matter, 
and  would  be  almost  exact. 

§  145.     Serial  Bonds 

Bonds  are  often  issued  in  series  so  that  they  mature  at 
various  dates.  For  example,  there  may  be  an  issue  of 
$30,000.00,  of  which  $1,000.00  is  payable  after  one  year, 
another  $1,000.00  after  two  years,  and  so  on,  the  final 
$1,000.00  being  payable  after  thirty  years.  Other  series  are 
more  complex,  as,  for  example,  $2,000.00  payable  each 
year  for  five  years,  and  $4,000.00  each  year  thereafter  for 
ten  years.  The  initial  value  of  a  series  on  any  given  basis 
cannot  be  found  by  one  operation ;  the  initial  value  of  each 
instalment  must  first  be  found,  and  the  sum  of  these  separate 
initial  values  gives  the  initial  value  of  the  entire  series. 
After  the  aggregate  initial  value  has  been  ascertained,  it 
may,  for  the  purposes  of  deriving  values  at  succeeding  in- 
terest periods,  be  treated  as  a  unit,  as  if  the  bonds  were 
not  in  series.  At  the  end  of  each  of  the  yearly  periods,  the 
ordinary  amortization  or  accumulation  would  have  to  be 
computed,  and  it  would  also  be  necessary  to  deduct  from  the 
total  value  the  par  value-  of  the  bonds  cancelled  or  retired. 

In  offering  serial  bonds  for  sale,  they  are  often  listed  as 
of  "average  maturity — 15^/2  years."  This  is  entirely  de- 
lusive, and  frequently  causes  the  buyer  to  believe  that  he  is 
getting  a  more  favorable  basis  than  will  be  realized.  The 
only  correct  valuation  of  a  series  is  the  sum  of  all  its  separate 
values.  If  we  assume  that  the  $30,000.00  above  referred  to 
was  a  series  of  5%  bonds  bought  on  an  assumed  3.50% 

basis,  the  true  value  would  be $35,005.00 

whereas  the  value  for  the  "average  time,"  i.e., 

151/2  years,  would  be 35,348.22 


VALUATION  OF  BONDS 


109 


In  computing  the  rate  of  income  yielded  by  a  series,  the 
income  rate  corresponding  to  the  average  time  may  be  taken 
as  a  point  of  departure,  but  it  will  be  found  that  it  is  in- 
variably too  high.  For  example,  let  us  assume  that  the 
thirty  5%  serial  bonds  mentioned  above  were  purchased 
when  first  issued  at  116.68,  and  that  the  income  rate  thereon 
is  desired. 

Looking  in  Sprague's  "Extended  Bond  Tables,"  under 
the  5%  bonds  in  the  15%  year  column  (average  date),  we 

find  that  the  value  nearest  to  116.68  is $1,165,200.00 

which  is  at  a  yield  of  approximately  3.60%. 

If,  however,  we  take  off  on  an  adding  machine, 
the  value  of  a  5%  bond  due  in  30  years  at 

3.60%   $1,255,549.38 

the  value  of  a  5%  bond  due  in  29  years  at 

3.60%   1,250,705.95 

the  value  of  a  5%  bond  due  in  28  years  at 

3.60%   1,245,686.60 

and  so  on,  to  and  including  the  bond  due  in 
1  year,  we  shall  find  that  the  total  value  of 

the  30  bonds  is 34,531,390.28 

and  that  the  average  price  is  therefore  115.10. 

An  income  yield  of  3.55%  for  the  above  bonds  will  re- 
sult in  a  value  of  116.06,  while  a  yield  of  3.50%  will  dis- 
close a  value  of  116.68.  The  true  yield  on  these  bonds,  if 
bought  at  116.68,  is  therefore  not  3.60%,  as  seems  at  first 
apparent,  but  is  3.50%. 

§  146.     Irredeemable  Bonds 

Sometimes,  as  in  the  case  of  British  Consols,  there  is 
no  right  nor  obligation  of  redemption.  If  the  government 
wishes  to  pay  off  any  of  its  bonds,  it  has  to  buy  them  at  the 
market  price.  With  this  class  of  bonds,  there  is  no  question 
of  amortization;  the  investment  is  simply  a  perpetual  an- 


no  THE  MATHEMATICS  OF  INVESTMENT 

nuity.  The  cash  interest  is  all  revenue,  and  the  original  cost 
is  the  constant  book  value.  If  £100  of  4%  Consols  be 
bought  at  96,  the  income  is  £4  per  annum,  and  the  book 
value  is  £96.  Since  the  investment  of  £96  produces  £4  an- 
nually, the  rate  of  income  is  4  -j-  96,  or  4  1/6%. 

§  147.     Optional  Redemption 

Sometimes  the  issuer  of  a  bond  has  the  right  to  redeem 
at  a  certain  date  earlier  than  the  date  at  which  he  must 
redeem.  It  must  always  be  expected  that  this  right  will  be 
exercised  if  profitable  to  the  issuer;  hence,  to  be  conserva- 
tive, a  purchaser,  when  buying  this  class  of  bonds  at  a 
premium,  must  always  consider  them  as  maturing,  or  reach- 
ing par,  at  the  earlier  date.  On  the  other  hand,  bonds  of 
this  character  bought  at  a  discount  must  be  considered  as 
running  to  the  longer  date.  If  the  bonds  bought  at  a 
premium  run  to  the  very  latest  date,  or  if  the  bonds  bought 
at  a  discount  are  called  for  redemption  at  an  earlier  date 
than  was  anticipated  by  the  investor,  he  will,  in  either  case, 
receive  a  higher  yield  in  his  income  rate  than  he  would  have 
received  on  the  more  conservative  basis.  The  element  of 
chance  enters  in  here,  but,  to  be  safe,  the  purchaser  should 
always  consider  that  the  chances  will  go  against  him ;  he  will 
then  have  all  to  gain  and  nothing  to  lose. 

The  option  of  redemption  is  sometimes  attended  by  a 
premium.  For  example,  the  issuer  of  a  thirty-year  bond 
reserves  the  right  to  redeem  after  twenty  years  at  105. 
Where  bonds  are  bought  at  such  an  income  yield  that  after 
twenty  years  the  book  value  will  be  more  than  105,  the  right 
of  redemption  at  105  is  a  detriment  to  the  purchaser.  In 
such  a  case  as  this,  the  safe  and  conservative  purchaser 
should  buy  at  such  an  income  basis  as  will  bring  the  book 
value  at  105  or  below  at  the  end  of  twenty  years. 

There  is  also  a  form  of  bond  issue,  not  uncommon  in 


VALUATION  OF  BONDS  HI 

Europe,  where  a  certain  or  indefinite  number  of  bonds  is 
drawn  by  lot  each  year  for  the  purposes  of  retirement.  As 
these  bonds  are  usually  issued  at  a  discount,  those  which  are 
drawn  at  the  earlier  dates  are  the  more  profitable.  The  in- 
vestor, however,  in  estimating  his  income,  must  assume  that 
his  particular  bonds  will  be  among  the  last  ones  drawn.  If 
drawn  at  earlier  dates,  there  is  a  profit  exactly  the  same  as 
that  arising  from  a  sale  above  book  value. 

§  148.    Bonds  as  Trust  Fund  Investments 

A  bond  which  has  been  purchased  by  a  trustee  at  a 
premium  is  subject  to  amortization  in  the  absence  of  testa- 
mentary instructions  to  the  contrary.  The  trustee  has  no 
right  to  pay  over  the  full  cash  interest  to  the  life  tenant, 
because  he  must  keep  the  principal  intact  for  the  remainder 
man.  If,  for  example,  the  trustee  were  to  invest  $104,- 
491.29  in  a  5%  bond  having  five  years  to  run,  and  if  he 
were  to  pay  over  the  full  amount  of  the  coupons  to  the  life 
tenant  for  the  period  of  five  years,  the  fund  at  the  end  of 
the  period  would  simply  be  the  par  value  of  the  bonds, 
$100,000.00,  and  would  therefore  be  depleted  to  the  extent 
of  $4,491.29,  to  the  manifest  injury  of  the  remainder  man. 
Since  the  investment  is  on  a  4%  basis,  the  trustee  should  pay 
over  at  the  end  of  the  first  half-year  only  2%  of  $104,491.29 
(or  $2,089.83),.  and  not  21/2%  of  $100,000.00  (or  $2,- 
500.00).  He  then  has  $410.17  cash  to  reinvest,  and  the  fund, 
including  this,  is  still  $104,491.29.  It  may  be  dif^cult  to 
invest  the  $410.17  at  as  favorable  a  rate  as  the  bonds,  very 
small  and  very  large  amounts  being  most  diflficult  to  invest. 
The  trustee  can  deposit  it  in  a  trust  company,  at  least,  and 
receive  interest  at  some  rate,  however  small. 

§  149.    Payments  to  Life  Tenant 

At  the  end  of  the  second  half-year,  the  net  income  on 


112  THE  MATHEMATICS  OF  INVESTMENT 

the  bond  is  only  $2,081.62;  but,  in  addition  to  this  amount, 
the  life  tenant  is  also  entitled  to  the  interest  on  the  $410.17. 
If  this  has  been  reinvested  at  exactly  4%  (interest  payable 
semi-annually),  the  interest  thereon  is  $8.20,  and  the  total 
amount  payable  to  the  life  tenant  is  $2,081.62 +  $8.20  = 
$2,089.82.  This  is  practically  the  same  amount  of  income 
as  in  the  first  half-year.  The  difference  between  the  coupons 
received  ($2,500.00)  and  the  net  income  ($2,089.82)  is  the 
amortization  ($418.38),  which  is  deposited  or  invested  as 
before.    The  trustee  now  has  in  the  fund : 

Book  value  of  the  bonds ,. . ..  .$103,662.74 

Invested  in  Trust  Company  or  otherwise  at  end 

of  first  half-year , ,. .  410.17 

Invested  in  Trust  Company  or  otherwise  at  end 

of  second  half-year 418.38 

Total $104,491.29 


He  has  paid  over  all  of  the  new  interest  earned,  and  he 
has  kept  the  corpus  or  principal  intact. 

§  150.     Effect  of  Varying  Rates  on  Investments 

Suppose,  however,  that  the  trustee  was  not  able  to  get 
4%  on  the  $410.17,  but  only  3%,  so  that  from  this  source 
would  come  only  $6.15,  making  the  total  income  $2,081.62 
plus  $6.15,  or  $2,087.77.  This  shows  a  slight  falling  off  in 
income,  but  that  is  to  be  expected  when  part  of  an  investment 
is  returned  and  reinvested  at  a  lower  rate.  If  the  reinvest- 
ment had  been  at  41/2%,  the  income  would  have  been  $2,- 
090.85,  slightly  more  than  the  first  half-year,  owing  to  the 
improved  demand  for  capital.  It  might  be  urged  that  the 
life  tenant  ought  to  receive  $2,089.83  semi-annually — no 
more,  no  less — being  at  a  4%  rate  on  $104,491.29.  This 
would  leave  $410.17  each  half-year  to  be  invested  in  a 


VALUATION  OF  BONDS 


"3 


Sinking  fund,  from  which  no  interest  should  be  drawn,  but 
which  should  be  left  to  accumulate  to  maturity,  when  it 
would  exactly  replace  the  premium,  if  compounded  at  4%, 
But  this  hope  might  not  be  realized.  Very  likely  the  average 
rate  would  be  less  or  more  than  4%  and  not  exactly  4%. 
If  less,  the  original  fund  would  be  to  some  extent  depleted, 
and  the  remainder  man  wronged;  if  more,  there  would  be 
too  much  in  the  fund,  and  the  life  tenant  would  receive  too 
little.  It  seems,  therefore,  that  the  sinking  fund  principle 
is  not  correct  in  a  case  like  this,  and  that,  at  all  events,  the 
original  fund  should  be  kept  constant,  neither  increased  nor 
diminished.  So  much  of  the  semi-annual  receipts  as  are  not 
necessary  to  maintain  the  constancy  of  the  fund  due  to  the 
remainder  man  should  be  paid  over  as  income  to  the  life 
tenant. 

§  151.    Example  of  Payments  to  Life  Tenant 

A  two-year,  4%  bond,  par  value  $10,000,  bought  at 
$10,192.72,  would  be  scheduled  thus : 


Coupon 

Income 

Cash 

Bond 

$10,192.72 

$200.00 

$162.89 

$47.11 

10,146.61 

200.00 

152.18 

47.82 

10,097.79 

200.00 

151.47 

48.63 

10,049.26 

200.00 

150.74 

49.26 

10,000.00 

$800.00 

$607.28 

$192.72 

The  life  tenant  would  receive,  at  the  end  of  the  first  half- 
year,  $152.89;  at  the  end  of  the  second,  $152.18  +  what- 
ever the  $47.11  cash  had  earned;  at  the  end  of  the  third, 
$151.47  +  whatever  $94.93  had  earned;  at  the  close, 
$150.74  -f  whatever  $143.46  had  earned.     If  the  cash  bal- 


114 


THE  MATHEMATICS  OF  INVESTMENT 


ance  were  periodically  deposited  in  a  trust  company  at  3% 
(payable  semi-annually),  the  life  tenant  would  receive  a  uni- 
form income  of  $152.89. 

§  152.    Cullen  Decision 

In  a  New  York  case  (38  App.  Div.  419),  Justice  Cullen 
very  clearly  lays  down  the  law  as  to  the  duty  of  the  trustee 
to  reserve  a  part  of  the  interest  to  provide  for  the  premium, 
and  says  that  "any  other  view  would  lead  to  the  impair- 
ment of  the  principal  of  the  trust,  to  protect  the  integrity 
of  which  has  always  been  the  cardinal  rule  of  courts  of 
equity."  He  says  further :  *'If  one  buys  a  ten-year  five  per 
cent  bond  at  one  hundred  and  twenty,  the  true  income  or 
interest  the  bond  pays  is  not  4  1/6%  on  the  amount  in- 
vested, nor  5%  on  the  face  of  the  bond,  but  2  7/10%  on 
the  investment,  or  3  24/100%  on  the  face  of  the  bond.  The 
matter  is  simply  one  of  arithmetical  calculation,  and 
tables  are  readily  accessible,  showing  the  result  of  the 
computation." 

§  153.    Cullen  Decision  Scheduled 

Consulting  one  of  the  tables  referred  to  by  Justice  Cullen 
(Sprague's  "Extended  Bond  Tables"),  and  looking  in  the 
5%  tables  under  the  column  headed  "10  Years,"  we  find 
that  the  value  nearest  to  $120,000.00  is  $120,038,997, 
which  is  opposite  the  net  income  rate  of  2.70%.  As  stated 
by  Justice  Cullen,  the  bonds,  while  bearing  a  coupon  rate  of 
5%,  actually  net,  therefore,  only  2.7%  on  account  of  the 
high  premium.  With  a  slight  correction  in  the  initial  figures 
in  order  to  make  the  income  rate  exactly  2.7%,  and  assum- 
ing a  par  value  of  $100,000.00,  the  illustration  as  given  in 
the  above  case  by  Justice  Cullen,  when  tabulated  to  show 
the  present  value,  the  income,  and  the  amount  reinvested, 
would  work  out  as  follows : 


VALUATION  OF  BONDS 


115 


Total 
Interest 

Income 
Paid  Over 

Reinvested 

Present 

Value 

$120,039.00 

$2,500.00 

$1,620.53 

$879.47 

119,159.53 

2,500.00 

1,608.65 

891.35 

118,268.18 

2,500.00 

1,596.62 

903.38 

117,364.80 

2,500.00 

1,584.42 

915.58 

116,449.22 

2,500.00 

1,572.07 

927.93 

115,521.29 

2,500.00 

1,559.53 

940.47 

114,580.82 

2,500.00 

1,546.85 

953.15 

113,627.67 

2,500.00 

1,533.97 

966.03 

112,661.64 

2,500.00 

1,520.93 

979.07 

111,682.57 

2,500.00 

1,507.72 

992.28 

110,690.29 

2,500.00 

1,494.31 

1,005.69 

109,684.60 

2,500.00 

1,480.75 

1,019.25 

108,665.35 

2,500.00 

1,466.93 

1,033.07 

107,632.28 

2,500.00 

1,453.08 

1,046.92 

106,585.36 

2,500.00 

1,438.91 

1,061.09 

105,524.27 

2,500.00 

1,424.57 

1,075.43 

104,448.84 

2,500.00 

1,410.06 

1,089.94 

103,358.90 

2,500.00 

1,395.35 

1,104.65 

102,254.25 

2,500.00 

1,380.43 

1,119.57 

101,134.68 

2,500.00 

1,365.32 

1,134.68 

100,000.00 

$50,000.00 

$29,961.00 

$20,039.00 

§  154.     Unjust  Feature  of  Cullen  Decision 

The  foregoing  schedule  is  perfectly  correct,  but  we  can 
scarcely  agree  with  the  method  described  further  on  in  the 
same  opinion,  as  follows:  "There  is,  however,  a  simpler 
way  of  preserving  the  principal  intact — the  method  adopted 
by  the  learned  referee.  He  divided  the  premium  paid  for 
the  bonds  by  the  number  of  interest  payments,  which  would 


ii6 


THE  MATHEMATICS  OF  INVESTMENT 


be  made  up  to  the  maturity  of  the  bonds,  and  held  that  the 
quotient  should  be  deducted  from  each  interest  payment 
and  held  as  principal.  These  deductions  being  principal, 
the  life  tenant  would  get  the  benefit  of  any  interest  that 
they  might  earn.  We  do  not  see  why  this  plan  does  not 
work  equal  justice  between  the  parties."  The  reason  "why 
it  does  not  work  equal  justice"  is  that  the  life  tenant  in  the 
earlier  years  receives  much  less  than  his  due  share  of  the 
income,  but  from  year  to  year  he  gradually  receives  more 
and  more,  until  he  receives  more  than  his  share;  but  not 
until  the  very  last  payment  does  he  overtake  his  true  share. 
Thus,  if  he  dies  before  the  maturity  of  the  bonds,  it  is  cer- 
tain that  "equal  justice"  will  not  have  been  done,  and  that 
the  remainder  man  will  have  had  altogether  the  best  of  it. 
The  schedule  under  the  referee's  plan  would  work  out  as 
follows : 


Total 
Interest 

Income 
Paid  Over 

Reinvested 

Present 
Value 

$120,039.00 

$2,500.00 

$1,498.05 

$1,001.95 

119,037.05 

2,600.00 

1,498.05 

1,001.95 

118,035.10 

2,500.00 

1,498.05 

1,001.95 

117,033.15 

2,500.00 

1,498.05 

1,001.95 

116,031.20 

2,500.00 

1,498.05 

1,001.95 

115,029.25 

2,500.00 

1,498.05 

1,001.95 

114,027.30 

2,500.00 

1,498.05 

1,001.95 

113,025.35 

2,500.00 

1,498.05 

1,001.95 

112,023.40 

2,500.00 

1,498.05 

1,001.95 

111,021.45 

3,500.00 

1,498.05 

1,001.95 

110,019.50 

etc. 

etc. 

etc. 

etc. 

Assuming  that,  under  each  plan,  the  reinvested  funds 
would  earn  the  same  rate  of  income  as  the  oriynal  invest- 


VALUATION  OF  BONDS  II7 

ment  (i.e.,  2.7%),  the  total  semi-annual  income  of  the  re- 
mainder man  would  be  as  follows : 


Under  Plan  Under  Plan 

in  §  153  in  §  154 

$1,620.53  $1,498.05 

1,620.53  1,511.58 

1,620.53  1,525.10 
1,620.53                                .         1,538.63 

1,620.53  1,552.16 

1,620.53  1,565.68 

1,620.53  1,579.21 

1,620.53  1,592.73 

1,620.53  1,606.26 

1,620.53  1,619.79 

etc.  etc. 

A  comparison  of  the  two  columns  will  show  the  injus- 
tice of  the  referee's  plan  toward  the  life  tenant,  and  sub- 
stantiate the  equity  of  the  plan  of  scientific  amortization  set 
forth  in  the  schedule  in  §  153. 

§  155.     Bond  Tables 

Mention  has  been  made  heretofore  of  bond  tables.  These 
tables  show  the  values  of  bonds  at  various  coupon  rates, 
yielding  various  rates  of  net  income,  and  due  in  different 
periods  from  one-half  year  to  one  hundred  years.  The  usual 
tables  refer  to  bonds  whose  coupons  are  payable  semi- 
annually, but  there  are  generally  supplementary  tables  by 
the  use  of  which  values  of  those  bonds  may  be  ascertained 
whose  coupons  are  payable  annually  or  quarterly.  The  fol- 
lowing table  is  taken  from  page  80  of  Sprague's  "Extended 
Bond  Tables,"  and  sets  forth  (in  part)  the  values  of  the 
bond  mentioned  in  Schedule  (A),  §122: 


ii8 


THE  MATHEMATICS  OF  INVESTMENT 


Bond  Table 

Values,  to  the  Nearest  Cent,  of  a  Bond  for  $1,000,000  at 
6%  Interest,  Payable  Semi-Annually 


Net 
In- 
come 


3  Years 


3J^  Years 


4  Years 


4^  Years 


5  Years 


2.50 
2.55 
2.60 
2.65 
2.70 
2.75 
2.80 
2.85 
2.90 
2.95 
3.00 
3.05 
3.10 
3.15 
3.20 
3.25 
3.30 
3.35 
3.40 
3.45 
3.50 
3.55 
3.60 
3.65 
3.70 
3.75 
3.80 
3.85 
3.90 
3.95 
4.00 
4.05 
4.10 
4.15 
4.20 
4.2S 


,071,825.12 
,070,328.46 
,068,834.33 
,067,342.73 
,065,853.65 
,064,367.09 
,062,883.04 
,061,401.50 
,059,922.46 
,058,445.92 
,056,971.87 
,055,500.31 
,054,031.24 
,052,564.64 
,051,100.52 
,049,638.87 
,048,179.68 
,046,722.96 
,045,268.68 
,043,816.86 
,042,367.48 
,040,920.54 
,039,476.04 
,038,033.97 
,036,594.33 
,035,157.11 
,033,722.30 
,032,289.91 
,030,859.92 
,029,432.34 
,028,007.15 
,026,584.36 
,025,163.96 
,023,745.94 
,022,330.31 
,020,917.04 


1,083,284.07 
1,081,538.84 
1,079,796.97 
1,078,058.45 
1,076,323.28 
1,074,591.45 
1,072,862.96 
1,071,137.78 
1,069,415.93 
1,067,697.38 
1,065,982.14 
1,064,270.19 
1,062,561.54 
1,060,856.16 
1,059,154.06 
1,057,455.22 
1,055,759.65 
1,054,067.33 
1,052,378.25 
1,050,692.42 
1,049,009.81 
1,047,330.43 
1,045,654.27 
1,043,981.31 
1,042,311.57 
1,040,645.01 
1,038,981.65 
1,037,321.47 
1,035,664.46 
1,034,010.63 
1,032,359.96 
1,030,712.44 
1,029,068.07 
1,027,426.84 
1,025,788.74 
1,024,153.77 


1,094,601.55 
1,092,608.09 
1,090,618.92 
1,088,634.05 
1,086,653.46 
1,084,677.14 
1,082,705.08 
1,080,737.28 
1,078,773.71 
1,076,814.37 
1,074,859.25 
1,072,908.34 
1,070,961.63 
1,069,019.11 
1,067,080.77 
1,065,146.59 
1,063,216.58 
1,061,290.71 
1,059,368.98 
1,057,451.38 
1,055,537.90 
1,053,628.52 
1,051,723.25 
1,049,822.06 
1,047,924.95 
1,046,031.91 
1,044,142.93 
1,042,258.00 
1,040,377.11 
1,038,500.25 
1,036,627.41 
1,034,758.58 
1,032,893.74 
1,031,032.90 
1,029,176.04 
1,027,323.16 


1,105,779.31 
1,103,537.98 
1,101,302.00 
1,099,071.36 
1,096,846.04 
1,094,626.03 
1,092,411.33 
1,090,201.90 
1,087,997.74 
1,085,798.84 
1,083,605.17 
1,081,416.74 
1,079,233.51 
1,077,055.49 
1,074,882.64 
1,072,714.97 
1,070,552.46 
1,068,395.09 
1,066,242.85 
1,064,095.73 
1,061,953.71 
1,059,816.78 
1,057,684.92 
1,055,558.13 
1,053,436.38 
1,051,319.67 
1,049,207.98 
1,047,101.30 
1,044,999.62 
1,042,902.92 
1,040,811.18 
1,038,724.41 
1,036,642.57 
1,034,565.67 
1,032,493.68 
1,030,426.59 


1,116,819.07 
1,114,330.27 
1,111,847.97 
1,109,372.18 
1,106,902.85 
1,104,439.98 
1,101,983.56 
1,099,533.55 
1,097,089.94 
1,094,652.71 
1,092,221.85 
1,089,797.33 
1,087,379.13 
1,084,967.25 
1,082,561.66 
1,080,162.34 
1,077,769.27 
1,075,382.44 
1,073,001.82 
1,070,627.41 
1,068,259.17 
1,065,897.10 
1,063,541.18 
1,061,191.38 
1,058,847.70 
1,056,510.11 
1,054,178.59 
1,051,853.13 
1,049,533.71 
1,047,220.32 
1,044,912.93 
1,042,611.52 
1,040,316.09 
1,038,026.61 
1,035,743.07 
1,033,465.45 


VALUATION  OF  BONDS 


119 


Net 
In- 
come 

3  Years 

354  Years 

4  Years 

454  Years 

5  Years 

4.30 

1,019,506.15 

1,022,521.93 

1,025,474.23 

1,028,364.40 

1,031,193.73 

4.35 

1,018,097.62 

1,020,893.20 

1,023,629.26 

1,026,307.08 

1,028,927.90 

4.40 

1,016,691.46 

1,019,267.57 

1,021,788.23 

1,024,254.63 

1,026,667.93 

4.45 

1,015,287.65 

1,017,645.05 

1,019,951.13 

1,022,207.03 

1,024,413.82 

4.50 

1.013,886.19 

1,016,025.62 

1,018,117.96 

1,020,164.27 

1,022,165.54 

4.55 

1,012,487.08 

1,014,409.27 

1,016,288.70 

1,018,126.33 

1,019,923.08 

4.60 

1,011,090.32 

1,012,796.01 

1,014,463.35 

1,016,093.21 

1,017,686.42 

4.65 

1,009,695.89 

1,011,185.82 

1,012,641.90 

1,014,064.89 

1,015,455.55 

4.70 

1,008,303.80 

1,009,578.70 

1,010,824.33 

1,012,041.35 

1,013,230.44 

4.75 

1,006,914.03 

1,007,974.64 

1,009,010.63 

1,010,022.60 

1,011,011.08 

4.80 

1,005,526.59 

1,006,373.63 

1,007,200.81 

1,008,008.60 

1,008,797.46 

4.85 

1,004,141.48 

1,004,775.67 

1,005,394.84 

1,005,999.36 

1,006,589.56 

4.90 

1,002,758.67 

1,003,180.75 

1,003,592.72. 

1,003,994.85 

1,004,387.36 

4.95 

1,001,378.18 

1,001,588.86 

1,001,794.45 

1,001,995.07 

1,002,190.85 

5.00 

1,000,000.00 

1,000,000.00 

1,000,000.00 

1,000,000.00 

1,000,000.00 

§  156.    Features  of  the  Bond  Table 

The  relations  existing  between  succeeding  values  on  the 
same  horizontal  line  of  the  table  are  readily  seen,  these 
values  being  computed  at  the  same  income  rate  but  for 
different  semi-annual  periods.  For  example,  on  a  2.50% 
basis,  the  value  of  this  5%  bond  5  years  prior  to  maturity  is 
$1,116,819.07.  At  4%  years  prior  to  maturity,  its  value  is 
1.0125  (which  is  the  semi-annual  income  rate)  times  $1,- 
116,819.07,  producing  $1,130,779.31,  from  which  must  be 
deducted  the  semi-annual  coupons  ($25,000.00),  giving  as 
a  final  result  $1,105,779.31.-  The  net  income  yields  shown 
in  the  preceding  table  vary  to  the  extent  of  .05%.  Usually 
there  are  supplementary  tables  by  the  use  of  which  values 
of  bonds  may  be  calculated  at  different  income  yields  differ- 
ing by  only  .01%.  In  this  manner,  the  values  of  the  bond 
shown  in  the  preceding  table  may  be  computed  on  an  income 
yield  of  2.51%,  2.52%,  2.53%,  etc. 


CHAPTER  XII 

SUMMARY  OF  COMPOUND  INTEREST 
PROCESSES 

§  157.    Rules  and  Formulas 

In  the  present  chapter  are  given  in  condensed  and  sym- 
bolic form  the  rules  and  formulas  which  have  been  explained 
in  the  preceding  chapters. 

§  158.    Rules 

(1)  To  find  the  ratio  of  increase : 

Add  1  to  the  rate  of  interest. 

(2)  To  find  the  amount  of  $1 : 

Multiply  1  by  the  ratio  as  many  times  as  there 
are  periods. 

(3)  To  find  the  present  worth  of  $1,  or  to  discount  $1 : 

Divide  1  by  the  ratio  as  many  times  as  there 
are  periods. 

(4)  To  find  the  total  interest  on  $1 : 

Subtract  1  from  the  amount. 

(5)  To  find  the  total  discount  on  $1 : 

Subtract  the  present  worth  from  1. 

(6)  To  find  the  amount  of  an  annuity  of  $1 : 

Divide  the  total  interest  by  the  rate  of  interest. 

(7)  To  find  the  present  worth  of  an  annuity  of  $1 : 

Divide  the  total  discount  by  the  rate  of  interest. 
120 


COMPOUND  INTEREST  PROCESSES  12 1 

(8)  To  find  the  rent  of  an  annuity  worth  $1,  or  what 

annuity  can  be  bought  for  $1 : 

Divide  1  by  the  present  worth  of  the  annuity. 

(9)  To  find  what  annuity  (sinking  fund)  will  produce 

$1: 

Divide  1  by  the  amount  of  the  annuity, 

(10)  To  find  the  premium  on  a  bond,  or  the  discount  on 

a  bond : 

Consider  the  difference  between  the  cash  and 
income  rates  as  an  annuity  to  be  valued,  and 
find  its  present  worth  at  the  income  rate. 

( 11 )  To  find  the  value  of  a  bond : 

In  case  the  cash  rate  is  greater  than  the  income 
rate,  the  bond  is  at  a  premium;  therefore, 
add  par  to  the  premium. 

In  case  the  income  rate  is  greater  than  the  cash 
rate,  the  bond  is  at  a  discount;  therefore, 
subtract  the  discount  from  par. 


§  159.     Formulas 

i  is  the  rate  of  interest  or  the  interest  on  unity  for  1 
period,  n  is  the  number  of  periods,  c  is  the  cash  rate  on  a 
bond. 


(1)   Ratio  of  Increase           = 

1  +  i 

(2)  Amount                            = 

(1+ir 

(3)   Present  Worth  &i  ^C-"^**^ 

1 

(i+ir 

(4)  Total  Interest                 = 

(i+ir-i 

(5)  Total  Discount               = 

1-    1 

122  THE  MATHEMATICS  OF  INVESTMENT 

(6)  Amount  of  Annuity         = 


(7)  Present  Worth  of  An- 
nuity 


(8)  Rent  of  Annuity 

(9)  Sinking  Fund 


i 
1 


X  — 

(1+ir 

i 
i 

1- 

1 

i 

(l+i)**-! 

(10)  Premium  on  Bond  =         — : — (  1  —   .^    ,    ..  „  ) 

(11)  Discount  on  Bond  =         — ; — ( 1  —    .^    ,    ..„  ) 

(12)  Value  of  Bond  (at  a  ^^    ,   c-j  /    _         1       \ 

Premium)  i      \         (l  +  i)**/ 

(13)  Value  of  Bond  (at  a  ^          i-  c  /     _        1       \ 

Discount)*  ^          *      \         (l  +  iy) 


•  This  formula  is  equivalent  to  Formula  (12). 


CHAPTER  XIII 

ACCOUNTS— GENERAL   PRINCIPLES 

§  1 60.    Relation  of  General  Ledger  to  Subordinate  Ledgers 

In  any  extensive  system  of  accountancy,  in  order  to 
fulfill  the  opposite  requirements  of  minuteness  and  compre- 
hensiveness, it  is  necessary  to  keep,  in  some  form,  a  general 
ledger  and  various  subordinate  ledgers.  Each  account  in 
the  general  ledger,  as  a  rule,  comprises  or  summarizes  the 
entire  contents  of  one  subordinate  ledger.  Each  account  of 
the  general  ledger  comprises  groups  of  similar  accounts, 
which  are  handled  in  the  subordinate  ledgers  as  individual 
assets  or  as  groups  which  may  be  treated  as  individuals.  It 
is  the  province  of  the  general  ledger  to  give  information 
in  grand  totals  as  an  indicator  of  tendencies;  while  the 
function  of  the  subordinate  ledger  is  to  give  all  desired  in- 
formation as  to  details,  even  beyond  the  figures  required 
for  balancing — facts  not  only  of  numerical  accountancy,  but 
descriptive,  cautionary,  or  auxiliary.  Thus  the  general 
ledger  may  contain  a  ''Mortgages"  account,  which  will  show 
the  increase  or  decrease  of  the  amount  invested  on  mortgage, 
and  the  resultant  or  present  amount;  the  mortgage  ledger 
will  contain  an  account  for  each  separate  mortgage,  with 
additional  information  as  to  interest,  taxes,  insurance,  title, 
ownership,  security,  valuation,  or  any  other  thing  useful 
or  necessary  to  be  known. 

§  161.    The  Interest  Account 

We  shall  assume  that  a  general  ledger  exists  with  subor- 
dinate or  class  ledgers.     We  shall  also  assume  that  the 

123 


124  THE  MATHEMATICS  OF  INVESTMENT 

accounts  are  to  be  so  arranged  as  to  give  currently  the 
amount  of  interest  earned  up  to  any  time,  and  the  amount 
outstanding  and  overdue  at  any  time.  It  would  hardly 
seem  necessary  to  argue  this  point,  were  it  not  that  many 
large  investors  pay  no  attention  to  interest  until  it  matures, 
and  some  do  not  carry  it  into  their  accounts  until  it  is  paid. 
They  are  compelled  to  make  an  adjustment  on  their 
periodical  balancing  dates  "in  the  air,"  compiling  it  from 
various  sources  without  check,  which  seems  as  crude  as  it 
would  be  to  take  no  account  of  cash,  except  by  counting  it 
occasionally.  The  Profit  and  Loss  account  depends  for  its 
accuracy  upon  the  interest  earned,  not  upon  the  interest 
falling  due,  nor  upon  the  interest  collected ;  and  the  accruing 
of  interest  is  a  fact  which  should  be  recognized  and  recorded. 

§  162.    Mortgage  and  Loan  Accounts 

In  considering  the  forms  of  account  for  investments,  we 
will  first  take  up,  as  being  simpler,  those  in  which  there  is 
never  any  value  to  be  considered  other  than  par,  such  as 
mortgages  and  loans  upon  collateral  security.  Both  of  these 
classes  of  investments  are  for  comparatively  short  terms, 
and  are  usually  the  result  of  direct  negotiation  between 
borrower  and  lender,  and  not  the  subject  of  purchase  and 
sale;  hence,  changes  in  rate  of  interest  are  readily  effected 
by  agreement,  and  do  not  result  in  a  premium  or  discount. 


CHAPTER  XIV 

REAL   ESTATE   MORTGAGES 

§163.    Nature  of  Loans  on  Bond  and  Mortgage 

The  instruments  which  we  have  spoken  of  as  "bonds" 
are  very  often  secured  by  a  mortgage  of  property.  But  one 
mortgage  will  secure  a  great  number  of  bonds,  the  mort- 
gagee being  a  trustee  for  all  the  bondholders.  In  contrast 
with  these  instruments,  those  of  which  we  now  speak  are 
the  ordinary  *'bond  and  mortgage,"  by  which  the  investor 
receives  from  the  borrower  two  instruments :  the  one  an 
agreement  to  pay,  and  the  other  conferring  the  right,  in 
case  default  is  made,  to  have  certain  real  estate  sold,  and 
the  proceeds  used  to  pay  the  debt.  As  only  a  portion  of  the 
value  of  the  real  estate  is  loaned,  the  reliance  is  primarily 
on  the  mortgage  rather  than  on  the  bond.  Therefore,  the 
mortgagee  must  be  vigilant  in  seeing  that  his  margin  is 
not  reduced  to  a  hazardous  point.  This  may  happen  by  the 
depreciation  of  the  land  for  various  economic  reasons;  by 
the  deterioration  of  the  structures  thereon  through  time  or 
neglect;  by  destruction  through  fire;  or  by  the  non-payment 
of  taxes,  which  are  a  lien  superior  to  all  mortgages.  By 
reason  of  these  risks  a  mortgage  loan  is  seldom  made  for 
more  than  a  few  years ;  but  after  the  date  of  maturity,  ex- 
tensions are  made  from  time  to  time;  or,  even  more  fre- 
quently, without  formal  extension,  the  loan  is  allowed  to 
remain  "on  demand,"  either  party  having  the  right  to  ter- 
minate the  relation  at  will.  A  large  proportion  of  outstand- 
ing mortgages  are  thus  "on  sufferance,"  or  payable  on 

125 


126  THE  MATHEMATICS  OF  INVESTMENT 

demand.  The  market  rate  of  interest  seldom  causes  the 
obHgation  to  change  hands  at  either  a  premium  or  a  dis- 
count; hence  we  may  ignore  that  feature,  referring  the 
exceptional  cases,  where  it  occurs,  to  the  analogy  of  bonds. 
The  two  instruments,  bond  and  mortgage,  relate  to  the 
same  transaction,  are  held  by  the  same  owner,  and  for  most 
purposes  are  treated  as  a  unit.  In  bookkeeping,  the  invest- 
ment must  likewise  be  treated  as  a  unit,  both  as  to  principal 
and  income. 

§  164.     Separate  Accounts  for  Principal  and  Interest 

It  is  desirable  to  know  at  any  time  how  much  is  due  on 
principal,  allowing  for  any  partial  payments.  It  is  also  de- 
sirable to  know  what  interest,  if  any,  is  due  and  payable,  and 
to  be  able  to  look  after  its  collection.  An  account  with 
principal  and  an  account  with  interest  are  therefore  requisite. 
It  is  better,  however,  if  these  two  accounts  for  the  same 
mortgage  be  adjacent. 

§  165.     Interest  Debits  and  Credits 

Accrued  interest  need  not  be  considered  as  to  each 
mortgage.  It  should  be  treated  in  bulk,  in  the  same  manner 
as  the  revenue  of  the  aggregate  mortgages,  as  will  be  ex- 
plained hereafter.  The  Interest  account  (on  the  investor's 
books)  here  referred  to  is  debited  on  the  day  when  the  inter- 
est becomes  a  matured  obligation,  and  credited  when  that 
obligation  is  discharged. 

§  166.     Characteristics  of  Modern  Ledger 

Those  who  adhere  to  the  original  form  of  the  Italian 
ledger  will  probably  be  averse  to  combining  with  the  ledger 
account  any  general  business  information ;  in  fact,  that  form 
is  not  suited  for  such  purposes,  and  is  not  adapted  to  con- 
taining anything  but  the  bare  figures  that  will  make  the 


J  J 


REAL  ESTATE  MORTGAGES 


127 


trial  balance  prove.  But  the  modern  conception  of  a  ledger 
is  broader  and  more  practical :  it  should  be  an  encyclopedia 
of  information  bearing  on  the  subject  of  the  account;  it 
should  be  specialized  for  every  class  ledger ;  it  should  be  of 
any  form  which  will  best  serve  its  purposes,  regardless  of 
custom  or  tradition. 

§  167.     The  Mortgage  Ledger 

The  form  of  mortgage  ledger  which  seems  best  to  the 
author  contains  four  parts : 

(1)  Descriptive. 

(2)  Account  with  principal. 

(3)  Account  with  interest. 

(4)  Auxiliary  information. 

These  may  occupy  four  successive  pages,  or  two  pages,  if 
preferred.  In  the  latter  case,  if  kept  in  a  bound  volume,  the 
arrangement  whereby  two  of  these  parts  are  on  the  left- 
hand  page  and  two  on  the  right,  confronting  each  other,  is 
a  convenient  one,  giving  all  the  facts  at  one  view.  For  a 
loose-leaf  ledger,  the  order  (1),  (2),  (3),  (4)  will  generally 
be  found  the  best. 

§  168.     Identification  of  Mortgages  by  Number 

Mortgages  should  be  numbered  in  chronological  order, 
and  every  page  or  document  should  bear  the  number  of  the 
mortgage  loan  to  which  it  refers. 

§  169.     The  "Principal"  Account 

The  account  with  principal  may  be  in  the  ordinary 
ledger  form;  but  what  is  known  as  the  balance-column,  or 
three-column  form,  will  be  found  more  convenient.  It  con- 
tains but  one  date  column,  so  that  successive  transactions, 
whether  payments  on  account,  or  additional  sums  loaned, 
appear  in  their  proper  chronological  order. 


128  THE  MATHEMATICS  OF  INVESTMENT 

§  170.     Special  Columns  for  Mortgagee's  Disbursements 

The  mortgage  usually  contains  clauses  which  permit  the 
mortgagee,  when  the  mortgagor  fails  to  make  any  necessary 
payment  for  the  benefit  of  the  property,  like  taxes  and  in- 
surance premiums,  to  step  in  and  advance  the  money,  which 
he  has  the  right  to  recover  with  interest.  It  will  be  use- 
ful to  have  columns  for  these  disbursements  and  the  corre- 
sponding reimbursements  (§  179,  Form  II). 

§  171.     The  Interest  Account 

The  Interest  account  (§  179,  Form  III)  may  be  very 
simple.  It  contains  two  columns,  one  for  debits  on  the  day 
when  interest  falls  due,  the  other  for  crediting  when  it  is 
collected.  The  entries  in  the  Interest  account  will  naturally 
be  much  more  numerous  than  in  the  Principal  account; 
hence,  this  pair  of  columns  may  be  repeated  several  times. 
The  arrangement  shown  has  been  found  advantageous. 

§  172.     Interest  Due 

Experience  has  shown  that  the  safest  way  to  insure  at- 
tention to  the  punctual  and  accurate  collection  of  interest  is 
to  charge  up,  systematically,  under  the  due  date,  every  item, 
and  to  let  it  stand  as  a  debit  balance  until  collected.  Many 
attempt  to  accomplish  the  same  purpose  by  merely  marking 
"paid"  on  a  list;  but  this  is  apt  to  lead  to  confusion,  and 
it  is  difficult  to  verify  afterward  the  state  of  the  accounts 
on  any  given  date. 

§  173.     Books  Auxiliary  to  Ledger 

It  is  not  proposed  in.  this  treatise  to  prescribe  the 
forms  of  posting  mediums  (cash  book,  journal,  etc.)  from 
which  the  postings  in  the  ledger  are  made,  because  these 
forms  are  so  largely  dependent  upon  the  peculiarities  of  the 


REAL  ESTATE  MORTGAGES  1 29 

business,  and  have  deviated  so  far  from  the  traditional 
Italian  form,  that  no  universal  type  could  be  presented.  We 
shall,  however,  give  the  debit  and  credit  formulas  underlying 
the  postings,  and  will  suggest  auxiliary  books  or  lists  for 
making  up  the  entries. 

§  174.     The  "Due"  Column 

The  formula  for  the  "Due"  column  of  the  Interest 
account  is : 

Interest  Due  /  Interest  Accrued        $ 

It  is  a  transfer  from  one  branch  of  interest  receivable,  viz., 
that  which  is  a  debt,  but  not  yet  enforceable,  to  another 
branch,  viz.,  that  which  is  a  matured  claim. 

§  175.     Interest  Account  Must  Be  Analyzed 

In  the  general  ledger  the  entry  will  be  simply  as  above : 

Interest  Due  /  Interest  Accrued        $ 

and  this  may  be  a  daily,  weekly,  or  monthly  entry,  or  for 
any  other  space  of  time,  according  to  the  general  practice 
of  the  business ;  the  monthly  period  is  most  in  use,  and  we 
shall  take  that  as  the  standard.  The  credit  side  of  the  entry 
(/  Interest  Accrued)  is  not  regarded  in  the  subordinate 
ledger  (§160),  but  the  debit  entry  (Interest  Due  /)  must 
be  somewhere  analyzed  into  its  component  parts;  in  other 
words,  there  must  be  somewhere  a  list,  the  total  of  which  is 
the  aggregate  falling  due  on  all  mortgages,  and  the  items 
of  which  are  the  interest  falling  due  on  each  mortgage. 

§  176.     Form  of  "Interest  Due"  Account 

The  following  heading  will  suggest  the  requirements 
for  such  a  list,  the  form  to  be  modified  to  conform  to  the 
general  system. 


130  THE  MATHEMATICS  OF  INVESTMENT 

Register  of  Interest  Due 
Mortgages 


Date 


No. 


Principal  Rate 


Time  Interest  Total 


§  177.     Forms  for  Mortgage  Account 

Form  I  (§  179)  of  the  Mortgage  account  is  descriptive. 
Its  elements  may  be  placed  in  various  orders  of  arrange- 
ment. Form  IV  (§  179)  combines  all  the  particulars  or- 
dinarily required  in  the  State  of  New  York. 

§  178.     Loose-Leaf  and  Card  Records 

Form  IV  (§  179)  is  not  an  essential  feature  of  mortgage 
loan  accounts,  and  may  be  replaced  by  card  lists,  if  pre- 
ferred. Yet,  if  there  is  space,  there  are  advantages  in  hav- 
ing all  the  information  about  a  certain  mortgage  accessible 
at  one  time,  and  concentrated  in  one  place.  The  changing 
names  and  addresses  of  the  mortgagors  and  owners,  and  the 
successive  policies  of  insurance  require  for  their  record 
considerable  space,  which  may  be  conveniently  arranged 
under  the  headings  in  Form  IV. 

The  card  form  of  mortgage  ledger  is  very  convenient 
in  many  respects,  and  the  forms  here  given  may  be  re- 
arranged to  suit  different  sizes  of  cards.  Both  in  cards  and 
loose  leaves  it  will  be  helpful  to  use  different  colors  for 
pages  of  different  contents.  Where  interest  on  different 
mortgages  fall  due  in  different  months,  tags  marked  "J  J," 
"F  A,'*  "M  S,"  "A  O,"  "M  N,"  and  "J  D,"  may  project 
from  the  interest-sheet  like  an  index,  the  tags  of  each 
month  at  the  same  distance  from  the  top.  This  will  greatly 
facilitate  the  compiling  of  the  register  of  interest  due. 


REAL  ESTATE  MORTGAGES 
§  179.    Forms  of  Mortgage  Loan  Accounts 


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REAL  ESTATE  MORTGAGES 


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THE  MATHEMATICS  OF  INVESTMENT 


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REAL  ESTATE  MORTGAGES 


135 


§  180.     Reverse  Posting  of  Interest  Register 

The  interest  register  should  always  be  made  up  and 
proved  (subjected  to  modifications)  in  advance.  In  doing 
this,  instead  of  making  the  computations  in  the  register  and 
posting  thence  to  the  ledger,  a  surer  way  is  by  "reverse 
posting" ;  that  is,  making  the  computation  from  the  data  in 
the  ledger  and  entering  it  there  at  once,  in  pencil  if  pre- 
ferred; then  copying  the  items  into  the  register,  where  the 
total  can  be  proved.  When  this  has  been  done,  we  can  be 
sure  without  further  check  that  the  ledger  is  correct.  (See 
also  §183.) 

§  181.     Handling  Receipts  and  Notices 

It  is  desirable,  also,  to  have  receipts  prepared  in  advance 
ready  for  signature.  The  correctness  of  these  receipts  may 
be  assured  by  introducing  them  into  the  "reverse  posting" 
process,  as  follows:  Having  made  the  computation  on  the 
ledger,  prepare  the  receipts  from  the  ledger,  copying  down 
the  figures  just  as  they  appear;  from  the  receipts  make  up 
the  register,  which  prove  as  before.  This  method  may  be 
extended  to  the  notices,  if  any  are  sent  to  the  mortgagors, 
the  notice  being  derived  from  ledger  account,  the  receipt 
from  the  notice,  and  the  register  from  the  receipt;  if  the 
register  proves  correct,  the  correctness  of  its  antecedents  is 
established.  These  interest  notices  may  be  made  of  as- 
sistance in  the  bookkeeping,  if  their  return  is  insisted  upon 
and  made  convenient.  Below  the  formal  notification  of  the 
sum  falling  due  on  such  a  date,  with  all  particulars,  is  a 
blank  form  somewhat  as  follows : 

In  payment  of  the  above  interest  I  inclose  check  on  the 

for  $ and 

request  you  to  acknowledge  receipt  as  per  the  address  below. 

Signature 

Address 


136 


THE  MATHEMATICS  OF  INVESTMENT 


The  notice  upon  its  being  received,  together  with  the 
check,  becomes  a  "voucher-with-cash,"  and  the  entries  on 
the  cash  book  and  the  interest  page  of  the  mortgage  ledger 
are  made  directly  from  the  documents.  Book-to-book  post- 
ing, which  formerly  was  the  only  method  of  rearranging 
items,  is  becoming  obsolete,  being  superseded  in  many  busi- 
nesses by  voucher  or  document  posting.  By  the  carbon 
process  the  notice  and  the  receipt  may  be  filled  in  simulta- 
neously in  fac-simile. 

§  182.     Mortgages  Account  in  General  Ledger 

The  class  account  "Mortgages"  in  the  general  ledger  is 
simply  kept  to  show  aggregates.  Its  entries  are,  as  far  as 
possible,  monthly,  the  posting  mediums  being  so  arranged 
as  to  give  a  monthly  total  of  the  same  items  which  have  al- 
ready been  posted  in  detail  to  the  mortgage  ledger.  The 
standard  form  of  ledger  account  may  be  used,  or  the  three 
column.  In  the  former,  the  debits  and  credits  of  the  same 
month  should  be  kept  in  line,  even  though  one  line  of  paper 
be  wasted. 


(Standard  Form) 


Mortgages 


1914 

1914 

Jan.            0 

Balance 

$169,000 

00 

1-31 

Total  loaned 

12,000 

00 

Jan. 

1-31 

Total  paid  in 

$     7,000 

00 

Feb.      1-28 

<(         « 

10,000 

00 

March  1-31 

«                  X 

50,000 

00 

March 

1-31 

«        «    <t 

32,000 

00 

April    1-30 

.. 

20,000 

00 

April 

1-30 

<<        <<    << 

40,000 

00 

May     1-31 

<l          « 

5,000 

00 

May 

1-31 

«        «    t< 

12,000 

00 

June     1-30 

<(          t< 

10,000 

00 

June 

1-30 

<t       <<    << 

3,000 

00 

Balance 

00 

30 

Balance 

182,000 

00 

$276,000 

$276,000 

00 

July           0 

$182,000 

00 

REAL  ESTATE  MORTGAGES 


137 


(Three-Column  Form) 


Mortgages 


Dk. 

Cr. 

Balance 

1914 
Jan.           0 

$169,000 

00 

" 

Transactions  for  month 

$   12,000 

00 

$    7,000 

00 

174,000 

00 

Feb. 

♦.             .t 

' 

10,000 

00 

184,000 

00 

March 

« 

' 

50,000 

00 

32,000 

00 

202,000 

00 

April 

«             « 

' 

20,000 

00 

40,000 

00 

182,000 

00 

May 

«             « 

' 

5,000 

00 

12,000 

00 

175,000 

00 

June 

Transactions  for  half-year 

10,000 

00 
00 

3,000 

00 
00 

182,000 

00 

$107,000 

$94,000 

+  $  13,000 

00 

July        0 

$182,000 

00 

§  183.     Tabular  Register 

In  order  to  keep  the  fullest  control  of  the  interest  accru- 
ing and  falling  due  periodically,  it  is  useful  to  keep  tabular 
registers,  classifying  the  mortgages,  first,  by  rates  of  in- 
terest; and  second,  by  the  months  in  which  the  interest 
comes  due.  Those  investors  who  require  all  interest  to  be 
paid  at  the  same  date  can  dispense  with  the  latter.  The  two 
presentations  or  developments  may  be  on  opposite  pages, 
both  proved  by  the  same  totals. 


Form  I 
Mortgages  Classified  by  Rates  of  Interest 


Date 

Total 

Changes 

3^% 

4% 

4H% 

5% 

6% 

1914 

Jan.O 

$169,000 
7,000 

262- 
984  + 

$11,000 

$43,000 
7,000 

$50,000 
12,000 

$60,000 

$5,000 

$162,000 
12,000 

Feb.O 

$174,000 

$11,000 

$36,000 

$62,000 

$60,000 

$5,000 

138 


THE  MATHEMATICS  OF  INVESTMENT 

Form  II 
Mortgages  Classified  by  Interest  Dates 


Date 

Total 

Changes 

JJ 

FA 

MS 

AG 

MN 

i          -J. 

JD 

1914 

Jan.O 

$169,000 
7,000 

262- 
984  + 

$23,000 

$30,000 
12,000 

$4,000 

$8,000 

$90,000 

$14,000 
7,000 

$162,000 
12,000 

Feb.O 

$174,000, 

$23,000 

$42,000 

$4,000 

$8,000 

$90,000 

$  7,000 

The  numbers  in  the  column  headed  "Changes"  are  the 
serial  numbers  of  the  respective  mortgages. 

§  184.     Equal  Instalment  Method 

Mortgages  payable  in  equal  instalments,  each  covering 
the  interest  and  part  of  the  principal,  present  no  special 
difficulty.  The  value  of  the  periodical  instalment  should 
first  be  ascertained,  as  shown  in  §  76 ;  then  it  should  be 
separated  by  means  of  a  schedule  into  "Interest  on  Balances" 
and  "Payments  on  Principal,"  down  to  the  final  payment. 


CHAPTER  XV 
LOANS    ON    COLLATERAL 

§  185.     Short  Time  Loans  on  Personal  Property 

Short  time  investments  are  often  made  upon  the  security 
or  pledge  of  bonds,  stocks,  goods,  or  other  personal  property 
valued  at  more  than  the  amount  of  the  loan.  Frequently 
these  are  payable  on  demand,  and  are  known  as  "call  loans." 
It  is  evident  that  the  rate  of  interest  may  be  readjusted 
every  day,  or  as  often  as  either  party  is  dissatisfied,  and, 
if  an  agreement  cannot  be  reached,  the  loan  will  be  paid 
off.  Hence,  neither  premium  nor  discount  will  occur  in  this 
kind  of  investment,  and,  as  in  the  case  of  mortgages,  we 
need  only  concern  ourselves  with  principal  (at  par)  and 
interest. 

§  186.    Forms  for  Loan  Accounts 

The  accountancy  of  loans  is  even  simpler  than  that  of 
mortgages,  and  it  is  only  necessary  to  give  two  forms,  one 
for  Principal  account  and  Interest  account,  and  the  other  for 
the  ''Register  of  Collateral"  (§  188).  The  latter  account,  at 
least,  is  often  kept  on  cards  or  on  envelopes,  and  there  is 
great  danger  of  the  history  becoming  confused  and  unin- 
telligible through  erasures  and  changes  in  the  amounts  of 
collateral,  when  substitutions  are  made.  When  part  of  a 
certain  security  is  withdrawn,  the'  entire  line  should  be 
ruled  out,  and  the  reduced  quantity  rewritten  on  a  new  line. 
When  a  card  becomes  at  all  complicated,  it  is  better  to  insert 
a  fresh  one,  rewriting  all  collateral,  but  keeping  the  former 
card  with  the  new  one  until  the  loan  is  entirely  liquidated. 

139 


I40  THE  MATHEMATICS  OF  INVESTMENT 

§  187.     Requirements  for  Interest  Account 

The  Interest  account  may  be  kept  concurrent  with  the 
Principal  account — that  is,  using  up  the  same  number  of 
lines  in  each.  In  the  suggested  form  there  is  a  column  for 
interest  accrued  as  well  as  for  interest  due.  The  interest 
accrued  column  is  merely  a  preparatory  calculation  column, 
entered  up  at  each  change  of  rate  or  principal,  so  that  there 
may  be  only  one  computation  to  make  when  the  interest 
becomes  due.  With  this  exception,  the  mechanism  of  the 
loan  ledger  is  practically  the  same  as  that  of  the  mortgage 
ledger,  and  the  general  ledger  account  of  loans  will  be 
similar  to  that  of  mortgages. 

As  the  principal  and  the  interest  in  bond  accounts  are 
so  intimately  connected,  it  will  be  advisable  to  consider  the 
accounting  of  interest  revenue  more  fully  before  taking 
up  the  subject  of  bond  accounts.  This  is  done  in  the  follow- 
ing chapter. 

§  188.     Forms  for  Collateral  Loan  Accounts 

On  the  following  page  are  shown  two  suggested  forms 
for  use  in  connection  with  the  records  for  collateral  loans. 
These  forms  are  merely  suggestive,  and  in  this  respect  are 
like  other  forms  presented  in  the  various  chapters  on  book- 
keeping records.  Very  few  banks  or  trust  companies  handle 
their  accounts  in  exactly  the  same  way,  and  changes  and 
additions  will  therefore  be  necessary  or  advisable  in  making 
use  of  the  forms  suggested,  in  order  to  meet  the  particular 
requirements  of  individual  companies : 


LOANS   ON    COLLATERAL 


141 


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CHAPTER  XVI 

INTEREST  ACCOUNTS 

§  189.     Functions  of  the  Three  Interest  Accounts 

Interest  is  earned  and  accrues  every  day;  then,  at  con- 
venient periods,  it  matures  and  becomes  collectible;  then  or 
thereafter  it  is  collected  and  takes  the  form  of  cash.  These 
three  stages  may  be  represented  by  the  bookkeeping 
formulas : 

(1)  Interest  Accrued  /  Interest  Revenue 

(2)  Interest  Due  /  Interest  Accrued 

(3)  Cash  /  Interest  Due 

Frequently  the  three  accounts,  Interest  Revenue,  Interest 
Accrued,  and  Interest  Due,  are  confused  under  the  one  title 
"Interest,"  although  they  have  three  distinct  functions.  In- 
terest Revenue  (which  alone  may  be  termed  simply  "Inter—, 
est")  shows  how  much  interest  has  been  earned  during  the 
current  fiscal  period.  The  balance  of  Interest  Accrued  shows 
how  much  of  those  earnings  and  of  the  earnings  in  previous 
periods  has  not  yet  fallen  due.  The  balance  of  Interest  Due 
shows  how  much  of  that  which  has  fallen  due  remains 
uncollected. 

The  first  of  the  three  entries  is  the  only  one  which  imports 
a  modification  in  the  wealth  of  the  proprietor;  the  other 
two  are  merely  permutative,  representing  a  shifting  from 
one  kind  of  asset  to  another.  It  is  not  the  mere  collecting 
of  interest  which  increases  wealth;  nor  is  it  merely  the 
coming-due  of  the  interest :  it  is  the  earning  of  it  from  day 
to  day. 

142 


INTEREST  ACCOUNTS  143 

§  190.    A  Double  Record  for  Interest  Earned 

Interest  accrued  need  not,  and  cannot  conveniently,  be 
computed  on  each  unit  of  investment,  as  we  have  already 
stated.  But  it  can  readily  be  computed  on  all  investments 
of  the  same  kind  and  rate  of  interest,  and  the  aggregate 
(say  for  a  month)  will  form  the  basis  for  the  entry  ''Interest 
Accrued  /  Interest  Revenue."  Or  a  daily  rate  for  the  entire 
investment  may  be  established,  and  this  may  be  used  without 
change,  day  after  day,  until  some  change  in  the  principal 
or  in  the  rate  causes  a  variation  in  the  daily  increment.  The 
most  complete  and  accurate  method  is  to  keep  a  double 
register  of  interest  earned :  first,  by  daily  additions ;  second, 
by  monthly  aggregates,  classified  under  rates  and  time. 

§  191.     Example  of  Interest  Income 

To  illustrate  this,  we  will  take  a  period  of  ten  days 
instead  of  a  month,  and  assume  that  the  investments  are  in 
mortgages  only.  On  the  first  day  of  the  period  there  is 
$100,000  running  at  4%,  $60,000  at  41/2%,  and  $150,000 
at  5%.  On  the  second  day,  $10,000  at  4%  is  paid  off,  and 
on  the  fifth  day  $5,000  at  5%.  On  the  seventh  day  a  loan 
of  $15,000  is  made  at  41/2%,  and  one  of  $6,000  at  5%.  We 
begin  by  computing  the  daily  increment,  as  follows : 

One  day  at  4%       on  $100,000 $11.1111 

One  day  at  41/2%   on  $  60,000 7.50 

One  day  at  5%       on  $150,000 20.8333 

Total  daily  increment $39.4444 

§  192.     Daily  Register  of  Interest  Accruing 

The  decimals  are  carried  out  two  places  beyond  the 
cents,  and  rounded  only  in  the  total.  The  daily  register  will 
then  be  conducted  as  follows : 


144 


THE  MATHEMATICS  OF  INVESTMENT 


Daily  Register  of  Interest  Accruing 
For  the  month  of ,  1914 


Date 

No. 

of 

Loan 

Decrease  in 
Principal 

Increase  in 
Principal 

Rate 

Working 
Column 

Daily 
Increment 

1 
2 

3 

4 
5 

6 

7 

8 

9 

10 

647 

453 

981 

982 

$10,000 
5,000 

$  15,000 
6,000 

4 

5 

4^ 
5 

$39.4444 
1.1111 

$  39.4444 
39.4444 

38.3333 
38.3333 
38.3333 

37.6388 
37.6388 

40.3472 
40.3472 
40.3472 

$38.3333 
.6944 

$37.6388 
1.875 
.8333 

$15,000 

$  21,000 

$390.21 

Balances 

at  Close 

$  90,000 

75,000 

151,000 

4 

4H 

5 

Proof 
of  Rate 
One  day 

$  10. 
9.375 
20.9722 

$316,000 

$  40.3472 

> 

§  193.     Monthly  Summary 

The  monthly  register  or  summary  takes  up,  first,  the 
mortgages  upon  which  payments  are  made,  then  those  re- 
maining to  the  end  of  the  month,  whether  old  or  new.  Its 
result  will  corroborate  that  of  the  daily  register. 

The  monthly  register  or  summary  of  interest  accruing 
may  be  kept  in  the  following  form.  As  the  loans  are  paid 
off,  the  interest  accrued  is  entered  up  in  the  last  column. 
New  loans  negotiated,  or  increases  in  principal,  are  entered 
in  column  four,  and  the  interest  accruing  to  date  of  pay- 
ment is  carried  to  the  last  column  in  a  similar  manner. 


INTEREST  ACCOUNTS 

Monthly  Summary  of  Interest  Accruing 
For  the  month  of ,  1914 


145 


Date 

No. 
of  Loan 

PakioflE 

Remaining 

Rate 

Days 

Monthly 
Increment 

2 

5 

7 
« 

10 
<• 
«« 

647 
453 

981 
982 

$10,000 
5,000 

$  15,000 

6,000 

90,000 

60,000 

145,000 

4 

5 

4J^ 

5 

4 

5 

2 

5 

3 

3 

10 

10 

10 

$     2.2222 
3.4722 
5.625 
2.50 
100.00 
75.00 
201.3888 

$316,000 

$390.21 

§  194.     Method  and  Importance  of  Interest  Earned  Account 

The  daily  and  monthly  registers  of  interest  earned  may 
be  in  separate  books  or  in  one  book — ^preferably  the  latter 
in  most  cases.  A  convenient  arrangement  would  be  to  use 
two  confronting  pages  for  a  month,  one  and  one-half  pages 
for  the  daily  register,  and  one-half  page  for  the  monthly 
register.  If  an  accurate  daily  statement  of  affairs  is  kept, 
the  daily  interest  accrued  will  form  part  of  that  system. 
Again,  the  interest  on  mortgages,  on  bonds,  on  loans,  or  on 
discounts  may  be  separated  or  be  all  thrown  together.  In 
all  such  respects  the  individual  circumstances  must  govern, 
and  no  precise  forms  can  be  prescribed.  Our  main  conten- 
tion is  that  in  some  manner  interest  should  be  accounted  for 
when  earned  rather  than  when  collected,  or  when  due. 


§  195.    Interest  Accounts  in  General  Ledger 

The  general  ledger  accounts  of  Interest,  Interest  Ac- 
crued, and  Interest  Due  will  now  be  illustrated  in  simple 
form  as  to  mortgages  only.  It  is  easier  to  combine  the 
several  kinds  of  interest,  when  carrying  them  to  the  Profit 
and  Loss  account,  than  to  separate  them  if  they  are  all 
thrown  in  together  at  first. 


146 


THE  MATHEMATICS  OF  INVESTMENT 


Form  I — Interest  Revenue 
Mortgages 


1914 
June      30 


Carried  to  Profit 
and  Loss 


$4270 

60 
60 

$4270 

—    II 

1914 
Jan. 
Feb. 
March 
April 
May 
June 


1-31 
1-28 
1-31 
1-30 
1-31 
1-30 


Total  Earnings 


$  654 
708 
723 
756 
719 
708 


$4270 


Form  II — Interest  Accrued 
Mortgages 


1914 

1914 

Jan. 

0 

Balance 

$2362 

50 

1-31 

Earnings 

654 

58 

Jan.       1-31 

Due 

$1272 

50 

Feb. 

1-28 

" 

708 

25 

Feb.      1-28 

" 

125 

00 

March 

17 

Cash  for  Accrued 
on  No.  987 

58 

33 

1-31 

Earnings 

723 

34 

March  1-31 

" 

875 

00 

April 

1-30 

" 

756 

67 

April     1-30 

" 

625 

00 

May 

1-31 

" 

719 

44 

May      1-31 

1200 

00 

June 

1-30 

" 

708 

33 

June      1-30 

" 

65 

00 

0 

Balance 

44 

30 

Balance 

2528 

94 

$6691 

$6691 

44 

July 

$2528 

94 

Form  III — Interest  Due 
Mortgages 


1914 

1914 

Jan.         0 

Balance 

$  125 

00 

1-31 

Due 

1272 

50 

Jan.       1-31 

Collections 

$1325 

00 

Feb.      1-28 

" 

125 

00 

Feb.      1-28 

197 

50 

March  1-31 

« 

875 

00 

March  1-31 

850 

00 

April     1-30 

" 

625 

00 

April     1-30 

600 

00 

May      1-31 

« 

1200 

00 

May      1-31 

1200 

00 

June     1-30 

It 

65 

00 

June      1-30 

100 

00 

Balance 

50 

30 

Balance 

15 

00 

$4287 

$4287 

50 

July       0 

$     15  Ool 

INTEREST  ACCOUNTS 


147 


§  196.    Payment  of  Accrued  Interest 

There  is  one  entry  in  Interest  Accrued  account  which 
does  not  arise  from  earnings :  the  accrued  interest  on  Mort- 
gage No.  987,  which  is  paid  for  in  cash  on  March  17,  the 
mortgage  not  having  been  made  direct  with  the  mortgagor, 
but  purchased  from  a  previous  holder.  This  case  occurs 
frequently  in  bond  accounts,  but  not  so  often  in  connection 
with  mortgages. 


CHAPTER  XVII 

BONDS  AND  SIMILAR  SECURITIES 

§  197.    Investments  with  Fluctuating  Values 

The  investments  heretofore  considered  are  interest  bear- 
ing, but  bear  no  premium  nor  discount ;  the  variation  from 
time  to  time  is  in  the  rate  of  interest,  while  the  principal  is 
invariable.  When  we  consider  investments  whose  price  fluc- 
tuates, while  the  cash  rate  of  interest  is  constant,  the  problem 
is  more  difficult,  because  there  are  several  prices  which  it 
may  be  desired  to  record,  viz.,  the  original  cost,  the  market 
value,  the  par,  and  the  book  value  or  amortized  value.  The 
original  cost  and  the  par  are  the  extremes :  one  at  the  be- 
ginning, and  one  at  the  end  of  the  investment.  The  book 
values  are  intermediate  between  these,  and  represent  the 
investment  value,  falling  or  rising  to  par  by  a  regular  law, 
which  maintains  the  net  income  at  a  constant  rate.  The 
market  value  is  not  an  investment  value,  but  a  commercial 
one ;  it  is  the  price  at  which  the  investor  could  withdraw  his 
investment,  but  until  he  has  done  so,  he  has  not  profited  by 
its  rise,  nor  lost  by  its  fall.  So  long  as  he  retains  his 
investment,  the  market  value  does  not  affect  him,  nor  should 
it  enter  into  his  accounts.  It  is  valuable  information,  how- 
ever, from  time  to  time,  if  he  has  the  privilege  of  changing 
mvestments,  or  the  necessity  of  realizing. 

§  198.     Amortization  Account 

The  account  with  principal,  showing  at  each  half-year 
the  result  of  amortization,  is  very  suitably  kept  in  the  three- 

148 


BONDS  AND  SIMILAR  SECURITIES 


149 


column  or  balance-column  form  recommended  in  §  169  for 
mortgages.  Thus,  the  history  of  the  bonds  in  Schedule  (F), 
§  139,  would  be  thus  recorded  in  ledger  form : 


$100,000  Smithtown  5's  of  May  1,  19 

19 

Date 

Dr. 

Cr. 

Balance 

1914  May   1 
Nov.   1 

1915  May    1 
Nov.   1 

1916  May    1 

Purchased  from  A.  B.  &  Co. 
Amortization  (4%) 

$104,500 

$410.97 
419.19 
427.57 
436.12 

$104,089.03 
103,669.84 
103,212.27 
102,806.15 

§  199.     Effect  on  Schedule  of  Additional  Purchases 

In  case  of  an  additional  purchase  the  account  will,  of 
course,  be  debited  and  cash  credited.  It  will  then  be  neces- 
sary to  reconstruct  the  schedule  from  that  point  on.  This 
may  be  done  in  either  of  two  ways:  (1)  make  an  indepen- 
dent schedule  of  the  new  purchase,  and  then  consolidate  this 
with  the  old  one,  adding  the  terms;  or  (2)  add  together  the 
values  of  the  old  and  new  bonds  at  the  next  balance  date; 
find  what  the  basis  of  the  total  is,  eliminate  any  slight  resi- 
due (§§137  to  140,  inclusive),  and  proceed  with  the 
calculation.* 


§  200.    The  Bond  Sales  Account 

In  case  of  a  sale,  the  procedure  is  different.  Instead  of 
crediting  the  Bond  account  by  cash,  it  is  best  to  transfer  the 
amount  sold  to  a  Bond  Sales  account  at  its  book  value  com- 
puted down  to  the  day  of  sale ;  Bond  Sales  account  will  then 
show  a  debit,  and  the  cash  proceeds  will  be  credited  to  the 
same  account.  The  resultant  will  show  a  gain  or  loss  on  the 
sale,  and  at  the  balancing  date  the  account  will  be  closed  into 


*  Bonds  purchased  flat  should  be  separated  into  principal  and  interest. 


I50 


THE  MATHEMATICS  OF  INVESTMENT 


Profit  and  Loss.  Thus,  in  the  example  in  §  198,  we  will 
suppose  a  sale  on  August  1,  1916,  of  half  the  $100,000  at 
102.88,  or  $51,440.  We  find  the  book  value  of  the  $50,000 
on  August  1,  which  is  $51,291.86;  we  transfer  this  to  the 
debit  of  the  Bond  Sales  account  in  the  general  ledger,  which 
account  we  credit  with  the  $51,440  cash  proceeds.  Bond 
Sales  is  purely  a  Profit  and  Loss  account,  and  at  the  proper 
time  will  show  the  actual  profit  realized  on  the  sale,  $51,440 
—  $51,29L86  =  $148.14. 

Form  I — Bond  Ledger 
$100,000  Smithtown  5's  of  May  1,  1919 


Date 

Dr. 

Cr. 

Balance 

1914  May   1 

Purchased  from  A.  B.  &  Co. 

$104,500 

Nov.  1 

Amortization 

$      410.97 

$104,089.03 

1915  May   1 

«« 

419.19 

103,669.84 

Nov.  1 

" 

427.57 

103,242.27 

1916  May    1 

" 

436.12 

102,806.15 

Aug.  1 

Sale  to  C.  D.  &  Co. 

$50,000  @  102.88 

51,291.86 

51,514.29 

« 

Amortization  on  $50,000 

111.21 

51,403.08 

Nov.  1 

"            on  balance 

222.43 

51,180.65 

Form  II — General  Ledger 
Bond  Sales 


1914 
Aug.  1 


Smithtown  5's 


$51,291.86 


1914 
Aug  1 


Proceeds 


$51,440.00 


To  adjust  the  profit  in  the  Bond  account  itself  would  be 
as  unphilosophical  as  the  old-fashioned  Merchandise  account 
before  the  Purchases  and  Sales  accounts  were  introduced, 
and  even  more  awkward. 


BONDS  AND  SIMILAR  SECURITIES 


151 


§  201.     Requirements  as  to  Bond  Records 

Besides  the  book  value  of  a  bond,  the  par  is  also  needed 
because  the  cash  interest  is  reckoned  upon  the  par.  For 
some  purposes,  also,  it  is  useful  to  show  the  original  cost. 
We  must,  therefore,  provide  means  for  exhibiting  these  three 
values:  the  par,  the  original  cost,  and  the  book  value.  A 
mere  memorandum  of  par  and  cost  at  the  top  would  be 
sufficient  where  the  group  of  bonds  in  question  will  all  be 
held  to  the  same  date;  but  this  is  not  always  the  case,  and 
provision  must  be  made  for  increase  and  decrease.  The 
three-column  form  of  ledger  (§169),  constantly  exhibiting 
the  balance,  is  the  most  suitable  for  this  purpose  also.  But 
if  we  endeavor  to  display  all  of  these  forms  side  by  side,  we 
require  nine  columns,  and  this  makes  an  unwieldy  book. 
The  most  practical  way  is  to  abandon  the  use  of  debit  and 
credit  columns,  and  proceed  by  addition  and  subtraction,  or 
in  what  the  Italians  term  the  scalar  (ladder-like)  form, 
which  gives  a  perfectly  clear  result,  especially  if  the  balances 
are  all  written  in  red.  Headed  by  a  description  of  the  bonds, 
and  embracing,  also,  a  place  for  noting  the  market  value  at 
intervals  (not  as  matter  of  account,  but  of  information),  the 
Principal  account  will  appear  as  shown  in  Form  I  (page 
152). 

§  202.    Form  of  Bond  Ledger 

As  far  as  the  bond  ledger  is  concerned,  the  transfer  of 
the  $50,000  sold  to  Sales  account  is  final ;  we  have,  however, 
in  the  example  indicated  (§  200),  a  way  of  incorporating  a 
statement  of  the  profit  or  loss  in  the  margin  for  historical 
purposes.  The  amortization  of  November  1,  1916,  is  com- 
posed of  two  parts :  3  months  on  the  $50,000  sold,  $111.21; 
and  the  regular  6  months  on  the  $50,000  retained,  $222.43. 
In  the  example  given  in  §  200,  these  are  entered  separately; 
either  method  may  be  pursued,  but  on  the  whole  there  are 
greater  advantages  in  postponing  all  entries  of  amortization 


152 


THE  MATHEMATICS  OF  INVESTMENT 


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BONDS  AND  SIMILAR  SECURITIES 


153 


till  the  end  of  the  half-year.  The  three  months'  amortiza- 
tion of  the  bonds  sold  is  in  effect  implied  in  the  price  $51,- 
291.86,  which  is  reduced  by  the  amortization  ($111.21) 
from  $51,403.07,  the  half  of  $102,806.15,  but  it  need  not 
be  entered  till  November  1. 

§  203.     Interest  Due  Account 

The  register  of  interest  due  on  bonds  is  conducted  on 
precisely  the  same  principles  as  that  described  for  mortgages 
in  §  183 ;  in  fact,  they  are  but  subdivisions  of  the  same 
register.     Of  course,  only  the  cash  interest  is  considered. 

§204.    Interest  Account — Bond  Ledger 

The  interest  pages  of  the  bond  ledger  are  also  similar  to 
those  of  the  mortgage  ledger  (§  183),  but  the  dates  of  in- 
terest due  may  be  printed  in  advance,  there  being  but  little 
chance  of  partial  payments  disturbing  their  orderly  ar- 
rangement. 

The  paging  of  the  bond  ledger  will  probably  be  geo- 
graphical, as  far  as  possible,  in  respect  of  public  issues,  and 
alphabetical  in  respect  of  those  of  private  corporations.  The 
loose-leaf  plan  permits  an  indefinite  number  of  classifica- 
tions from  which  to  choose.  The  date  tags  suggested  in 
§  178  are  especially  useful  for  pointing  out  dates  for  interest 
falling  due,  as  "J  J,"  "F  A,"  etc. 

§  205.    Amortization  Entries 

The  entries  of  amortization  are  made  directly  from  the 
schedules  of  amortization,  the  preparation  of  which  was 
discussed  in  Chapter  X.  But  it  is  necessary,  also,  to  make 
up  a  list  of  these  several  amortizations  in  order  to  form  the 
general  ledger  entry : 

Amortization  /  Bonds 
or.  Amortization  /  Premiums 


154 


THE  MATHEMATICS  OF  INVESTMENT 


according  to  the  form  of  the  general  ledger.  This  list 
should  be  in  the  same  order  as  the  bond  ledger.  Probably 
the  most  practical  way  is  to  combine  it  with  the  trial  balance 
of  the  bond  ledger,  thus  giving  at  each  fiscal  period  a  com- 
plete list  of  the  holdings,  which  may  give  the  par,  cost,  book, 
and  market  values,  the  titles  of  the  securities  being  written 
but  once.  The  total  of  the  second  column  will  form  the  basis 
of  the  entry  for  amortization.  The  next  three  columns  will 
corroborate  the  general  ledger  balances. 

Bond  Statement  for  the  Half- Year  Ending 


Name  and 
Description 


Amorti- 
zation 


Book  Value 


Par  Value 


Original 
Cost 


Market 
Value 


In  the  cases  of  bonds  bought  at  a  discount,  the  analogous 
general  ledger  entry  would  be : 

Bonds  /  Accumulation 
or.  Discounts  /  Accumulation 

We  have  provided  in  the  preceding  form  for  amortization 
only  and  not  for  accumulation  on  bonds  below  par.  Where 
the  latter  values  are  few  in  number  they  may  be  embraced 
in  the  same  column,  but  distinguished  as  negatives  by  being 
written  in  red  or  encircled.  If  the  bonds  below  par  are 
numerous,  there  should  be  two  columns :  * 'amortization"  and 
"accumulation." 


§  206.     Bond  Entries  in  General  Ledger 

While  the  book  value  is  the  proper  one  to  be  introduced 


BONDS  AND  SIMILAR  SECURITIES  155 

into  the  general  ledger,  the  par  is  very  necessary,  and  some- 
times the  cost,  and  these  requirements  inevitably  introduce 
some  complexity.  There  are  two  methods  effecting  the 
purpose : 

(1)  By  considering  the  par  and  cost  as  extraneous  in- 

formation, and  ruling  side  columns  for  them 
beside  the  book  value. 

(2)  By  dividing  the  account  into  several  accounts,  by 

the  proper  combination  of  which  the  several 
values  may  be  obtained. 

The  first  plan  will  preserve  the  conformity  of  the  Bonds 
account  with  the  bond  ledger  better  than  the  other.  The 
Bonds  account  may,  if  necessary,  be  extended  across  both 
pages  of  the  ledger,  to  allow  for  three  debit  and  three  credit 
columns,  if  all  are  required. 

The  second  method  will  commend  itself  more  to  those 
having  a  repugnance  to  introducing  into  the  general  ledger 
any  figures  beyond  those  actually  forming  part  of  the  trial 
balance.  The  theory  on  which  it  is  based  is  that  the  premium 
is  not  part  of  the  bond,  but  is  a  sum  paid  in  advance  for 
excess  interest,  while  the  discount  is  a  rebate  returned  to 
make  good  deficient  interest.  This  is  a  perfectly  admissible 
way  of  looking  at  the  matter,  especially  from  the  personalis- 
tic  point  of  view;  for  the  debtor  does  not  owe  us  the 
premium  and  has  nothing  to  do  with  it.  Still  the  other  view, 
which  regards  the  investment  as  a  whole,  is  also  correct, 
and  wx  may  adopt  whichever  is  most  suitable  to  our  purpose. 

§  207.     Accounts  Where  Original  Cost  Is  Disregarded 

If  original  cost  is  disregarded,  or  deemed  easily  obtain- 
able when  required,  the  accounts  may  be : 


156  THE  MATHEMATICS  OF  INVESTMENT 

(a)  Bonds  at  Par 

(b)  Premiums 

(c)  Discounts 

or, 

(a)  Bonds  at  Par 

(b)  Premiums  and  Discounts 

If  premiums  and  discounts  are  kept  separate,  Premiums 
account  must  always  show  a  debit  balance,  being  credited 
for  amortization;  Discounts  account  must  show  a  credit 
balance,  being  debited  for  accumulation.  If  the  two  are 
consolidated,  only  the  net  amortization  will  be  credited 
(§  205)  ;  or  if  the  greater  part  of  the  bonds  were  below  par, 
the  net  accumulation  only  would  be  debited.  The  choice  be- 
tween one  account  and  two  for  premiums  and  discounts  is 
largely  a  question  of  convenience. 

The  management  of  such  a  double  or  triple  account  is 
obvious,  entries  of  transactions  being  divided  between  par 
and  premiums,  or  par  and  discounts,  but  we  give  in  §  214  an 
example  of  each. 

We  shall  hereafter  confine  the  discussion  to  premiums, 
leaving  the  cases  of  discount  to  be  determined  by  analogy. 

§  208.    Amortization  Reserve 

Where  it  is  deemed  necessary  to  keep  account  of  cost 
also,  as  well  as  of  par  and  book  value,  the  difficulty  is  some- 
what greater,  as  we  have  a  valueless  or  extinct  quantity  to 
record,  namely,  so  much  of  the  original  premium  on  bonds 
still  held  as  has  not  yet  been  absorbed  in  the  process  of 
amortization.  This  carrying  of  a  dead  value,  which  is  some- 
what artificial,  necessitates  the  carrying,  also,  of  an  artificial 
annulling  or  offsetting  account,  the  sole  function  of  which 
is  to  express  this  departed  value.    We  may  call  this  credit 


BONDS  AND  SIMILAR  SECURITIES 


157 


account  "Reserve  for  Amortization."  It  is  analogous  to 
Depreciation  and  Reserve  for  Depreciation.  The  part  of  the 
premiums  which  has  been  extinguished  bytcredits  to  Reserve 
for  Amortization  may  be  designated  as  "Premiums  Amor- 
tized," or  "Ineffective  Premiums,"  while  the  live  premiums 
may  be  styled  "Effective  Premiums,"  being  what  in  §  207 
we  called  simply  "Premiums."  A  double  operation  takes 
place  in  these  accounts :  first,  the  absorption  of  effective 
premiums  by  lapse  of  time;  and  second,  the  cancellation  of 
ineffective  premiums  upon  redemption  or  sale. 

§  209.     Premiums  and  Amortization 

There  are  two  ways  of  handling  these  accounts,  differ- 
ing as  to  premiums.  We  may  keep  two  accounts :  "Effec- 
tive Premiums"  and  "Amortized  Premiums,"  or  we  may 
combine  these  in  one,  "Premiums,  at  Cost."  The  entire 
scheme  will  be : 

(a)  Bonds  at  Par 

(b)  Premiums  at  Cost 

(e)   Reserve  for.  Amortization 

or, 

(a)  Bonds  at  Par 

(c)  Effective  Premiums 

(d)  Amortized  Premiums 

(e)   Reserve  for  Amortization 

"a"  will  in  both  schemes  be  the  same ;  "e"  will  also  be  the 
same,  "b"  is  the  sum  of  "c"  and  "d."  In  the  former,  the 
cost  is  a  +  b,  while  the  book  value  is  a  +  b  —  e.  In  the  latter 
the  book  value  is  a  +  c,  while  the  cost  is  a  +  c  +  d.  The 
former  gives  the  cost  more  readily  than  the  latter,  and  the 
book  value  less  readily.  The  former  might  be  considered  the 
more  suitable  for  a  trustee;  the  latter,  for  an  investor. 


158 


THE  MATHEMATICS  OF  INVESTMENT 


Account  (a),  Bonds  at  Par,  is  debited  for  par  value  of 
purchases  and  credited  for  par  value  of  sales.  Its  only  two 
entries  are : 

Bonds  at  Par  /  Cash  (or  some  other  asset) 
Cash  (or  some  other  asset)  /  Bonds  at  Par 

In  case  of  purchase  at  a  premium,  the  premium  is 
charged  to  Premiums  at  Cost  or  to  Effective  Premiums,  as 
the  case  may  be,  there  being  no  ineffective  premiums  at  this 
time. 

§  210.     Writing  Off  Premiums 

When  premiums  are  written  off,  on  the  first  plan  illus- 
trated in  §  209  there  is  but  one  entry :  crediting  Reserve  for 
Amortization  and  debiting  the  Profit  and  Loss  account  or 
its  subdivision. 

Amortization  /  Reserve  for  Amortization 

The  second  plan  involves  not  only  this  process,  but  a 
transfer  from  Effective  to  Amortized  Premiums.  Thus,  the 
aggregate  of  premiums  written  off  is  posted  four  times  as  a 
consequence  of  the  separation  of  premiums  at  cost  into  two 
accounts:  if 

Premiums  Amortized  /  Effective  Premiums 
Amortization  /  Reserve  for  Amortization 

§  211.     Disposal  of  Amortization 

The  word  "Amortization"  has  been  used  in  the  illustra- 
tive entries  as  the  title  of  an  account  tributary  to  Profit  and 
Loss.  At  the  balancing  period  it  may  be  disposed  of  in 
either  of  two  ways :  It  may  be  closed  into  Profit  and  Loss 
direct ;  or  it  may  be  closed  into  Interest  account,  the  balance 
of  which  will  enter  into  Profit  and  Loss  at  so  much  les- 


•BONDS  AND  SIMILAR  SECURITIES  icq 

sened  a  figure.  By  the  former  method  the  Profit  and  Loss 
account  will  show,  on  the  credit  side,  the  gross  cash  inter- 
est, and  on  the  debit  side  the  amount  devoted  to  amortiza- 
tion ;  the  second  method  exhibits  only  the  net  income  from 
interest  on  bonds.  Whether  it  be  preferable  to  show  both 
elements,  or  only  the  net  resultant,  will  be  determined  by 
expediency. 

§  212.    Amortization  Accounting — Comparison  of  Methods 

In  §§  200  and  202  we  discussed  two  methods  of  keeping 
account  of  amortization:  the  first  (in  §200),  where  any 
incidental  amortization  occurring  in  the  midst  of  the  period 
is  at  once  entered;  the  second  (in  §202),  where  all  such 
entries  are  deferred  to  the  end  of  the  period.,  and  comprised 
in  one  entry  in  the  general  ledger.  If  the  latter  method  be 
adopted,  the  Amortization  account  may  be  dispensed  with 
altogether,  and  the  total  amount  amortized  (which  is 
credited  to  Bonds,  or  to  Premiums,  or  to  Reserve  for 
Amortization)  may  be  debited  at  once  to  Profit  and  Loss  or 
to  Interest,  without  resting  in  a  special  account.  A  single 
item,  of  course,  needs  no  machinery  for  grouping. 

§  213.     Irredeemable  Bonds  a  Perpetual  Ai^nuity 

Irredeemable  bonds  (§  146)  merely  lack  the  element  of 
amortization,  and  require  no  special  arrangement  of  ac- 
counts. The  par  is  purely  ideal,  as  it  never  can  be  demanded 
and  is  merely  a  basis  for  expressing  the  interest  paid.  What 
the  investor  buys  is  a  perpetual  annuity.  If  he  buys  an 
annuity  of  $6  per  annum,  it  is  unimportant  whether  it  is 
called  6%  on  $100  principal,  or  4%  on  $150  principal;  and 
this  $150  may  be  the  par  value,  or  it  may  b.e  $100  par  at 
50%  premium,  or  $200  par  at  25%  discount.  The  par  value 
is  really  non-existent. 


l6o  THE  MATHEMATICS  OF  INVESTMENT 

§  214.     Bond  Accounts  for  General  Ledger 

In  the  present  section  are  shown  the  forms  for  the  gen- 
eral ledger  outlined  in  §§  206-212.  We  will  suppose  that 
on  January  1,  1915,  the  following  lots  of  bonds  are  held : 


January  1,  1915 
Par  Book  Value 

$100,000     5%  Bonds,  J  J, 

due  Jan.  1,  1925,  net  2.7%  ;  value.  .$120,039.00 
Original  cost,  $124,263.25 
100,000     3%  Bonds,  M  N, 

due  May  1,  1918,  net  4%;  value. .     96,909.10 
Original  cost,  $93,644.28 
10,000     4%  Bonds,  A  O, 

due  Oct.  1,  1916,  net  3%;  value..     10,169.19 
Original  cost,  $10,250.00 

$210,000  Totals  $227,117.29 


The  premiums  on  the  5%  and  4%  bonds  amount  to 
$20,208.19.  The  discount  on  the  3%  bonds  is  $3,090.90. 
The  net  premium  is  $17,117.29.  The  total  original  cost 
was  $228,157.53. 


BONDS  AND  SIMILAR  SECURITIES 


i6i 


o 


< 


W 

O 

« 
o 
P^ 

H 
O 

u 
u 

< 

o 


M> 


^  o 


o  o  th  eo  CO 

tT"   O  CI  O  C5 

d  d  \6  ci  d 

O   C   I-  lO  l-Ii 

-f   C   -^  r}<  Tf 
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O 

s.a 

I   o 


l62  THE  MATHEMATICS  OF  INVESTMENT 


Bond  Accounts  for  General  Ledger — Plan  II  (§  207) 
Dr.  Bonds  at  Par  Cr. 


1915 
Jan.  0, 


Balance $210,000.00 


1916 
Oct.  L 


Redeemed. $10,000.00 


Dr. 

Premiums 

Cr. 

1915 

1915 

Jan.  0, 

Balance $20,208.19 

June  30, 

Amortization. 

..$926.94 

Dec.  31, 

« 

..  939.54 

1916 

June  30, 

(( 

..  952.28 

Dec.  31, 

i( 

..  940.21 

1917 

June  30, 

u 

...  927.93 

Dec.  31, 

it 

..  940.47 

Dr. 

Discounts 

Cr. 

1915 

1915 

June  30, 

Accumulation.. . 

.$438.18 

Jan.  0, 

Balance $3,090.90 

Dec.  31, 

« 

.  446.95 

1916 

June  30, 

« 

.  455.88 

Dec.  31, 

it 

.  465.00 

1917 

i 

June  30. 

« 

.  474.30 

1 

Dec.  31. 

« 

.  483.79 

1 

BONDS  AND  SIMILAR  SECURITIES 


163 


Bond  Accounts  for  General  Ledger — Plan  III  (§  207) 
(Original  cost  omitted) 


Dr. 


Bonds  at  Par 


Cr, 


1915 
Jan.  0, 


Balance $210,000.00 


1916 
Oct.  1 


Redeemed $10,000.00 


Dr. 

Premiums  and  Discounts 

Cr. 

1915 
Jan.  0, 

Balance $17,117.29 

1915 
June  30, 
Dec.  31, 

Amortization. 

..$488.76 
..  492.59 

1916 
June  30, 
Dec.  31, 

u 
u 

..  496.40 
..  475.21 

1917 
June  30, 
Dec.  31, 

It 

..  453.63 
...  456.68 

164 


THE  MATHEMATICS  OF  INVESTMENT 


Bond  Accounts  for  General  Ledger  — Plan  IV  (§  209) 
("Bonds  at  Par"  as  in  foregoing  plans) 


Dr. 


Premiums  at  Cost 


Cr. 


1915 
Jan.  0, 


1918 
Jan.  0, 


Balance $18,157.53 


Balance, 


$18,157.53 
.$17,907.53 


1916 
Oct.  1, 


1917 
Dec.  31, 


Canceled  at  Re- 
demption... .$     250.00 


Balance 17.907.53 


$18,157.53 


Dr. 

Reserve  for 

Amortization                             Cr. 

1916 

1915 

Oct.  1, 

Canceled  at  Re- 

Jan.   0. 

Balance $1,040.24 

demption.. .  .$   250.00 

June  30, 

Amortization. . .      488.76 

Dec.  31, 

...      492.59 

1917 

Dec.  31, 

Balance 3,653.51 

1916 

June  30, 

...      496.40 

Dec.  31, 

...     475.21 

1917 

June  30, 

...     453.63 

Dec.  31, 

...     456.68 

$3,903.51 

$3,903.51 

1918 
Jan.     0, 

Balance $3,653.51 

BONDS  AND  SIMILAR  SECURITIES 


i6S 


Bond  Accounts  for  General  Ledger — Plan  V  (§  210) 
(By  the  balance  column  method) 


Bonds  at  Par 


Dr. 


Cr.       Balance  Dr. 


1915 
Jan.     0 

1916 
Oct.     1 


Balance 

Redemption. 


$210,000.00 


$10,000.00 


$210,000.00 
200,000.00 


Effective  Premiums 


Dr, 


Cr.       Balance  Dr. 


1915 
Jan.     0 
June  30 
Dec.  31 

1916 
June  30 
Dec.  31 

1917 
June  30 
Dec.  31 


Balance 

Amortized 


$  17,117.29 


$     488.76 
492.59 


496.40 
475.21 


453.63 
456.68 


$  17,117.29 
16,628.53 
16,135.94 

15,639.54 
15,164.33 

14,710.70 
14,254.02 


Ineffective  or  Amortized 
Premiums 


Dr. 


Cr.        Balance  Dr. 


1915 
Jan.     0 
June  30 
Dec.  31 

1916 
June  30 
Oct.     1 
Dec.  31 

1917 
June  30 
Dec.  31 


Balance 

Amortized 

« 

u 

Canceled  by  Redemption 
Amortized 


$  1,040.24 
488.76 
492.59 


496.40 


475.21 


453.63 
456.68 


$     250.00 


1,040.24 
1,529.00 
2,021.59 

2,517.99 
2,267.99 
2,743.20 

3,196.83 
3,653.51 


i66 


THE  MATHEMATICS  OF  INVESTMENT 


Reserve  for  Amortization 


Dr. 


Cr.        Balance  Cr. 


1915 
Jan.     0 
June  30 
Dec.  31 

1916 
June  30 
Oct.     1 
Dec.  31 

1917 
June  30 
Dec.  31 


Balance 

Amortized 

(( 

Canceled  by  Redemption 
Amortized 


$       250.00 


$  1,040.24 
488.76 
492.59 

496.40 
475.21 

453.63 
456.68 


1,040.24 
1,529.00 
2.021.59 


2,517.99 
2,267.99 
2,743.20 


3,196.83 
3.653.51 


CHAPTER  XVIII 

DISCOUNTED  VALUES 

§  215.     Securities  Payable  at  Fixed  Dates  Without  Interest 

The  securities  heretofore  considered  have  all  carried  a 
stipulated  rate  of  interest  or  annuity.  There  is  another  class 
to  which  no  periodical  interest  attaches,  but  the  obligation  is 
simply  to  pay  a  single  definite  sum  on  a  certain  date.  The 
present  vakie  of  that  sum  at  the  current  or  contractual  rate 
of  income  is,  of  course,  obtained  by  discounting  according 
to  the  principles  explained  in  Chapter  XL  If  the  maturity 
were  more  than  one  year  distant  at  the  time  of  discount,  it 
would  be  necessary  to  compute  the  compound  discount ;  but 
in  practice  this  never  occurs,  such  discounts  being  for  a  few 
months. 

The  obligations  discounted  in  this  manner  are  almost 
invariably  promissory  notes.  Formerly  they  consisted  large- 
ly of  bills  of  exchange;  hence  the  survival  in  bookkeeping 
of  the  words  "Bills  Receivable,"  "Bills  Payable,"  and  "Bills 
Discounted." 

These  obligations  belong  rather  to  mercantile  and  bank- 
ing accountancy  than  to  investment  accountancy.  The 
arrangement  of  accounts  for  recording  their  amounts,  classi- 
fication, and  maturity  has  been  so  fully  treated  in  works  on 
those  branches  that  we  refer  to  them  here  only  for  the 
purpose  of  illustrating  another  phase  of  the  process  of  secur- 
ing income. 

§  216.     Rates  of  Interest  and  Discount 

The  difference  between  the  rate  of  interest  and  the  rate 

167 


l68  THE  MATHEMATICS  OF  INVESTMENT 

of  discount  has  been  pointed  out  in  Chapter  II.  It  was 
there  shown  that  in  a  single  period  the  rate  of  interest  3% 
corresponds  to  the  rate  of  discount  .029126.  Hence,  if  we 
discount  a  note  for  $1.00  at  2.9126%,  we  acquire  interest  at 
the  rate  of  3%  on  the  $.970874  actually  invested.  The  rate 
of  interest  is  always  greater  than  the  rate  of  discount. 

§  217.     Rate  of  Discount  Named  in  Notes 

It  is  usual  to  name  a  rate  of  discount  rather  than  a  rate 
of  interest  in  stipulating  for  the  acquisition  of  notes.  For 
example,  a  three  months'  note  for  $1,000  is  taken  for  dis- 
count at  6%  (per  annum).  This  means  that  $.015  is  to  be 
retained  by  the  payee  of  the  note  from  each  dollar,  and  the 
amount  actually  paid  over  is  $985.  The  income  from  this 
is  the  $15,  and  by  dividing  15  by  985  we  readily  ascertain 
that  the  rate  of  interest  realized  is  6.09%.  It  is  sometimes 
believed  that  there  is  a  kind  of  deception  in  this;  that  the 
borrower  agrees  to  pay  6%  and  actually  has  to  pay  6.09%. 
But  this  is  not  so :  the  bargain  is  not  to  pay  6  %  interest,  but 
to  allow  6%  discount,  which  is  a  different  thing. 

§  218.     Form  as  Affecting  Legality 

Curiously,  the  lawfulness  or  unlawfulness  of  a  trans- 
action sometimes  depends  upon  the  mere  form  of  words  in 
which  it  is  expressed.  Thus,  suppose  that  A  lends  $985  to 
B,  who  promises  to  repay  $1,000  at  the  end  of  3  months.  If 
B's  promise  reads :  *T  promise  to  pay  $1,000,"  A  is  a  law- 
abiding  citizen;  but  if  B  writes :  "I  promise  to  pay  $985  and 
interest  at  6.09%  per  annum,"  the  statute  prohibiting  usury 
is  violated. 

§  219.    Entry  of  Notes  Discounted 

Notes  discounted  are  usually  entered  among  the  assets 
at  the  full  face,  and  the  discount  credited  to  an  offsetting 


DISCOUNTED  VALUES 


169 


account,  "Discounts,"  the  latter  having  precisely  the  same 
effect  as  the  Discounts  account  used  in  connection  with 
bonds.  The  difference  of  the  two  is  the  net  amount  of  the 
asset.  Strictly  speaking,  the  discount  is  at  first  an  offset  to 
the  note,  and  represents  at  that  time  nothing  earned  what- 
ever; as  time  goes  on,  the  earning  is  effected  by  diminution 
of  this  offset,  which  is  equivalent  to  a  rise  in  the  net  value 
of  the  note,  from  cost  to  par.  In  §  220  the  process  is  shown 
by  the  state  of  the  accounts  at  the  initial  date  and  at  the  end 
of  each  month  up  to  maturity,  for  a  3  months'  note  for 
$1,000,  discounted  at  6%. 


§  220.     Discount  and  Interest  Entries 


Note 


$1000.00 


(i)   When  Discounted 
Discount 


$15.00 


(2)  At  the  End  of  One  Month 
Note  Discount  Interest  Revenue 


$1000.00 


$  5.00 


$15.00 


$5.00 


(3)  At  the  End  of  Two  Months 
Note  Discount  Interest  Revenue 


$1000.00 


$  5.00 
5.00 


$15.00 


$5.00 
5.0U 


Note 


$1000.00 


(4)  At  Maturity 

Discount  Interest  Revenue 


$  5.00 
5.00 
5.00 

$15.00 

$15.00 

$15.00 

$5.00 
5.00 
5.00 


lyo  THE  MATHEMATICS  OF  INVESTMENT 

§  221.     Total  Earnings  from  Discounts 

Since  notes  are  issued  generally  for  short  periods,  the 
gradual  crediting  of  earnings  illustrated  in  §  220  is  usually 
ignored.  At  the  date  when  the  books  are  closed,  an  inven- 
tory should  be  taken  of  the  discounts  unearned.  The  differ- 
ence between  the  amount  of  this  inventory  and  the  net  credit 
in  the  Discounts  account  represents  the  earning  from  dis- 
counts during  the  fiscal  period,  and  this  earning  should  then 
be  transferred  to  Profit  and  Loss.  The  unearned  discounts 
may  be  easily  computed  by  finding  the  discount  on  each 
note  from  the  date  of  closing  the  books  to  the  respective 
dates  of  maturity.  The  investment  value  of  the  notes  on 
hand  at  the  close  of  the  fiscal  period  will  be  the  difference 
between  the  par  and  the  unearned  discount.  Expressed  in 
a  formula,  the  earnings  from  discounts  may  be  found  as 
follows : 

Unearned  discounts  at  beginning  of  fiscal  period, 
+        discounts  credited  during  period, 
—        unearned  discounts  at  end  of  period, 
=        earnings  from  discounts  during  period. 


Part  II — Problems  and  Studies 


CHAPTER  XIX 

INTEREST  AND  DISCOUNT 

§  222.     Problems  in  Simple  Interest* 

(1)  What  is  the  time  in  months  and  days  from  January 
10th  to : 

(a)  June  12th? 

(b)  July  4th? 

(c)  September  1st? 

(2)  What  date  is: 

(a)  Two  months  after  June  30th ? 

(b)  Four  months  after  May  31st? 

(c)  Two  months  after  December  31st? 

(d)  Five  months  and  seven  days  after  September 

26th? 

(3)  On  a  loan  of  $54,750,  interest  payable  semi-annually 
at  4%  per  annum,  interest  was  last  paid  to  and  including  No- 
vember 1 :  compute  the  interest  accrued  on  the  following 
February  25th : 

(a)  In  the  customary  manner,  legal  in  New  York 

before  1892. 

(b)  Assuming  that  the  odd  days  are  365ths  of  a 

year. 


•  In  connection  with  the  text  of  Chapter  II.     For  answers  see  §  224. 

171 


172 


PROBLEMS  AND  STUDIES 


(c)  Compute  the  same  by  both  methods  at  4%%. 

(d)  "  "       "      "      "  "        "  5%. 

(e)  "         "      "      "      "  "        "  6%. 

(4)  On  a  365-day  basis,  the  interest  for  17  days,  on  a 
certain  sum,  at  a  certain  rate,  was  $83.73 ;  what  would  have 
been  the  interest  on  a  360-day  basis  ? 

(5)  The  interest  for  19  days  on  a  certain  sum  at  a 
certain  rate  was  $2,185.00  on  a  360-day  basis;  compute 
the  interest  on  a  365-day  basis. 


§  223.     Notes  on  the  One  Per  Cent  Method 

Observe  that  when  days  are  considered  as  360ths  of  a 
year,  it  is  useful  to  know  how  many  days  correspond  to  one 
per  cent.  For  example,  if  the  rate  is  3%,  it  takes  120  in- 
terest days  to  earn  1%  interest. 

At  3     %  the  number  of  days  for  1%  is  120 


At  4  % 

((      (( 

"  90 

At  41/2% 

({      (f 

"  80 

At  5  % 

(t      (( 

"  72 

At  6  % 

((      (t 

"  60 

At  8  % 

(t      (( 

"  45 

At  9  % 

((      t( 

"  40 

For  purposes  of  calculation  we  may  set  down  the  num- 
ber of  days  corresponding  to  1%  at  the  given  rate,  and  in 
line  with  it  the  principal,  pointing  off  two  places  from  the 
right  in  the  principal  in  order  to  obtain  1%.  Thus,  in 
Problem  (3)  of  the  preceding  section : 
90  days  $547.50 
meaning  that  the  interest  for  90  days  at  4%  is  $547.50. 
Knowing  the  interest  for  90  days,  we  can  build  up  that  for 
114  days  (3  months  and  24  days  on  the  360-day  basis). 
24  days  =  15  days  +  9  days.    15  days  is  1/6  of  90  days;  9 


INTEREST  AND  DISCOUNT 


173 


days,  1/10.  Dividing  the  interest  for  90  days  by  6  to  secure 
the  interest  for  15  days,  and  by  10  to  secure  the  interest 
for  9  days,  and  adding,  gives  the  result : 

90  days    $547.50 

15     "  91.25 

9     "  54.Y5 


114     "       $693.50 


The  same  result  may  be  obtained,  and  just  as  easily,  by 
the  combination  90  +  18  +  6.  Sometimes  the  work  may  be 
shortened  by  the  use  of  subtraction ;  in  the  present  case,  no 
time  would  be  saved  by  this  method,  the  result  working  out 
as  follows : 

90  days  $547.50 

30     "  $182.50 

less     6  (1/5  of  30)     36.50       146.00 


114  $693.50 


In  the  case  of  problem  (3-d),  on  the  5%  basis,  the  result 
would  work  out  as  follows : 

72  days  $547.50 

18     "      (1/4  of  72)       136.875 
24     "      (Ys  of  72)       182.50 

114     "  $866,875 


Rates  like  7%  or  3%%,  which  are  not  exact  divisors  of 
360,  must  be  obtained  from  the  exact  rates  by  division  and 
addition.  Thus,  7%  is  derived  by  adding  1/6  to  6%,  which 
is  obtained  as  follows : 


ly^  PROBLEMS  AND  STUDIES 

60  days    $547.50 

30  "  273.75 

20  "  182.50 

4  "  36.50 


114     "     $1,040.25   (interest  at  Q%) 
add  1/6       173.375 


$1,213,625   (interest  at  7%) 

§  224.    Answers  to  Problems  in  Simple  Interest 

Problem   (1) 

(a)  5  months,  2  days 

(b)  5  months,  24  days 

(c)  7  months,  22  days 

Problem   (2) 

(a)  August  30th 

(b)  September  30th 

(c)  February  28th  or  29th 

(d)  March  4th  or  5th 

Problem  (3) 

(a)  $693.50 

(b)  $691.50 

(c)  $780.19   (360-day  method) 
$777.94  (365-day  method) 

(d)  $866,875   (360-day  method) 
$864,375    (365-day  method) 

(e)  $1,040.25   (360-day  method) 
$1,037.25   (365-day  method) 

Problem   (4) 

$84.89 
Problem   (5) 

$2,155.07 


INTEREST  AND  DISCOUNT 


175 


§  225.     Problems  in  Compound  Interest* 

(6)  Find  the  amount  of  $1  at  2%  per  period,  correct  to 
six  decimals : 

(a)  For  one  period 

(b)  For  two  periods 

(c)  For  three  periods 

(d)  For  four  periods 

(e)  For  five  periods 

(7)  Find  the  present  worth  of  $1  at  2%,  correct  to  six 
decimals : 

(a)  For  one  period 

(b)  For  two  periods 

(c)  For  three  periods 

(d)  For  four  periods 

(e)  For  five  periods 

In  §  29,  several  methods  are  mentioned  for  finding  the 
present  worth;  assuming  that  the  solutions  for  problem 
(6)  above  have  been  found,  the  easiest  method  of  finding 
the  present  worth  for  five  periods  would  be  to  divide  1  by 
the  amount  for  five  periods.  The  present  worths  for  1,  2, 
3,  and  4  periods  can  then  be  found  by  multiplying  the  present 
worth  for  five  periods  successively  by  1.03.'  This  is  much 
easier  than  dividing  1  successively  by  1.03. 

(8)  Find  the  amount  of  $1  at  1^^%  (.0176)  per 
period : 

(a)  For  one  period 

(b)  For  two  periods 

(c)  For  three  periods 

(d)  For  four  periods 

(e)  For  five  periods 

(f)  For  six  periods 


*  In  connection  with  the  text  of  Chapter  II,    For  answers  see  §  226. 


176 


PROBLEMS  AND  STUDIES 


(9)  Find  the  present  worth  of  $1  at  1.75%  per  period : 

(a)  For  one  period 

(b)  For  two  periods 

(c)  For  three  periods 

(d)  For  four  periods 

(e)  For  five  periods 

(f)  For  six  periods 

(10)  Find  the  amount  and  the  present  worth  of  $1,- 
000.00  for  eight  periods  at  1.5%  per  period. 

(11)  What  is  the  rate  of  discount  corresponding  to  2% 
interest  ? 

(12)  What  is  the  rate  of  interest  corresponding  to  the 
discount  rate  of  .0384615? 

(13)  Three  notes  for  $1,000.00  each,  due  (without  in- 
terest) at  three  months,  six  months,  and  one  year  respec- 
tively, are  discounted  at  6%  : 

(a)  If  the  proceeds  of  the  first  note  are  $985,  find 

the  equivalent  interest  rate. 

(b)  If  the  proceeds  of  the  second  note  are  $970, 

find  the  equivalent  interest  rate. 

(c)  If  the  proceeds  of  the  third  note  are  $940, 

find  the  equivalent  interest  rate. 

(14)  What  is  the  compound  interest  on  $1  for  five 
periods  at  2%  ? 

(15)  What  is  the  compound  discount  on  $1  for  four 
periods  at  2%  ? 

§  226.     Answers  to  Problems  in  Compound  Interest 
Problem  (6) 

(a)  $1.02 

(b)  $1.0404 

(c)  $1.061208 

(d)  $1.082432 

(e)  $1.104081 


INTEREST  AND  DISCOUNT 


177 


Problem  (Y) 

(a)  $.980392 

(b)  $.961169 

(c)  $.942322 

(d)  $.923845 

(e)  $.905731 

Problem  (8) 

(f)  $1.109702 

Problem  (9) 

(f)   $.901143 

Problem  (10) 

Amount,  $1,126.493 ;  present  worth,  $887,711 

Problem   (11) 

.0196078  (Observe  that  this  decimal  when  divided 
by  2%,  the  rate  of  interest,  gives  the  present 
worth  for  one  period,  .98039.  This  will  be  a 
test  for  all  similar  computations.) 

Problem  (12) 

4% 

Problem   (13) 

(a)  1.52284%       quarterly,      or       (nominally) 

6.09137%  annually 

(b)  3.09278%    semi-annually,    or    (nominally) 

6.18557%  annually 

(c)  6.38298%  annually 

Problem  (14) 
$.104081 

Problem   (15) 
$.076155 

§  227.     Proof    of   Amount    and    Present    Worth 

The  amount  and  the  present  worth  of  the  same  sum  for 


178 


PROBLEMS  AND  STUDIES 


the  same  time  and  rate  should,  when  multiplied  together, 
give  the  product  1, 

Problems  (6)  and  (7)  give  the  amount  of  $1  for  5 
periods  at  2%  as  $1.104081,  and' its  present  value  for  the 
same  time  and  rate  as  $.905731.  These  numbers  multiplied 
together  should  give  as  a  product,  unity.  Such  multiplica- 
tions of  decimal  numbers  are  best  performed  by  beginning 
at  the  left  of  the  multiplier. 


1.10408 

1 

.905731 

993672 

9 

5520 

405 

772 

8567 

33 

12243 

1 

104081 

1.000000  388211 
The  vertical  line  is  drawn  to  cut  off  the  figures  beyond 
the  6th  decimal,  which  have  no  utility  except  to  furnish  a 
carrying  amount   for  the   6th  figure.     They  may  be  dis- 
pensed with  by  using  contracted  multiplication. 

§  228.     Contracted  Multiplication 

In  this  process  the  subproducts  are  shortened  at  each 
step  by  one  figure,  taking  into  account,  however,  the  carry- 
ing amount  from  the  rejected  figures. 

1.104081 
.905731 


(first  6  figures  X  9)... 

.    993673 

(first  5  figures  X  0)  . . . 

0 

(first  4  figures  X  5)... 

5520 

(first  3  figures  X  7) 

773 

(first  2  figures  X  3)... 

33 

(first       figure     X  1)... 

1 

1.000000 


INTEREST  AND  DISCOUNT 


179 


Here  we  commence  to  multiply  by  9  at  the  sixth  figure, 
8 ;  the  product  would  be  72,  but  we  know  that  the  rejected 
1,  X  9,  would  make  the  product  nearer  73 ;  this  subproduct, 
therefore,  becomes  993673.  In  each  of  these  partial  prod- 
ucts the  last  retained  figure  is  slightly  increased,  if  neces- 
sary, by  mental  allowance  for  the  next  rejected  figure.  The 
last  figure  of  the  final  product  will,  even  then,  not  always  be 
exact,  but  may  vary  one  or  two  units  from  the  correct  prod- 
uct. In  all  multiplications  by  rounded  decimals,  there  is 
an  error,  small  it  is  true,  in  the  product;  this  final  error 
may  be  reduced  to  as  small  a  quantity  as  desired,  by  in- 
creasing the  decimal  places  in  the  factors  to  such  extent 
as  the  accuracy  of  the  work  may  require. 

It  sometimes  happens  in  contracted  multiplication  that 
you  "lose  your  place"  and  forget  at  what  figure  of  the 
multiplicand  to  begin  next.  This  may  be  overcome  by  tick- 
ing off  each  figure  as  you  have  done  with  it ;  or  by  repeating 
the  multiplier  figures  from  left  to  right  and  (at  the  same 
time)  the  multiplicand  figures  from  right  to  left.  In  the 
above  illustration  the  correlated  figures  would  be  9-8,  0-0, 
5-4,  7-0,  3-1,  and  1-1. 

§  229.     Problems  in  Use  of  Logarithms* 

The  following  problems  are  elementary  and  the  4-place 
table  given  in  §  43  may  be  used  in  their  solution. 

(16)  What  is  the  logarithm  of : 

(a)  3        (d)   1.8  (g)     .54 

(b)  30        (e)     .0018  (h)   1.03 

(c)  3,000        (f)   5.4 

(17)  Give  the  number  whose  logarithm  is: 

(a)  .1614      (c)   1.6474  (e)   3.6474 

(b)  2.3838      (d)   1.6474  (f)     .0212 

*  In  connection  with  the  text  of  Chapter  III.      For  answers  see  §  231. 


l8o  PROBLEMS  AND  STUDIES 

(18)  Find  the  logarithm  of : 

(a)   291.5        (b)  4.362  (c)     .027433 

(19)  Find  the  number  whose  logarithm  is : 

(a)   2.5849      (b)   1.38425         (c)  3.6931 

(20)  Prove  by  logarithms  that : 

(a)  9X8  =  72 

(b)  7X1.12  =  7.84 

(c)  .032X300  =  9.6 

(d)  .004X4000  =  16 

(21)  Show  by  logarithms  that: 

(a)  72 -f- 2.4  =  30 

(b)  12.5 -^  625  =  .02 

(c)  5.2^.04  =  130 

(22)  What  is  the  28th  power  of : 
(a)  1.02  (b)   1.04 

(23)  What  is  the  present  worth  of  $1  for  45  periods  at : 
(a)  3%  (b)  5% 

(24)  Find  by  logarithms  the  value  of  the  following : 
829  X  76.3  X  .0484  -v-  7.28  -^  25 

§  230.    Problems  Requiring  Use  of  More  Extended  Tables 
of  Logarithms* 

For  further  exercise  in  logarithmic  computations,  Prob- 
lems (14)  to  (18)  inclusive  should  again  be  worked  out, 
using  logarithms  to  the  limit  of  such  tables  as  may  be  at 
hand.  The  logarithms  of  all  of  the  ordinary  ratios  of  in- 
crease (1  +  i),  with  which  the  operation  always  begins,  will 
be  found  in  Part  III.  These  logarithms  have  been  com- 
puted to  fifteen  places  of  decimals. 

The  following  examples,  which  are  for  too  many  periods 
to  be  worked  out  arithmetically,  may  also  be  worked  by 


For  answers  see  §  231. 


INTEREST  AND  DISCOUNT  igi 

logarithms.  If  no  other  tables  are  available,  the  four-place 
tables  in  §  43  may  be  used,  although  these  tables  cannot  be 
relied  upon  to  bring  correct  results  to  as  many  decimal  places 
as  are  given  in  the  solutions. 

(25)   Find  the  amount  and  present  worth  of  $1  ; 

(a)  At  1.25%  for  30  periods 

(b)  At  1.70%  for  50  periods 

(c)  At  2.00%  for  10  periods 

(d)  At  2.40%  for  68  periods 

(e)  At  2.50%  for  70  periods 

§231.    Answers  to  Problems  in  Logarithms 
Problem  (16) 

(a)  .4771 

(b)  1.4771 

(c)  3.4771 

Problem  (17) 

(a)  1.45 

(b)  242 
Problem  (18) 

(a)  2.4647 
Problem  (19) 

(a)  384.5 
Problem  (20) 

(a)  log.  9  =  .9542 ;  log.  8  =  .9031.    The  sum  of 
these  two  logarithms  is  1.8573,  which  is 
the  logarithm  of  72.    Similarly  for  (b), 
(c),  and  (d). 
Problem  (21) 

(a)  log.  72  =  1.8573 ;  log.  2.4  =  .3802.  The  first 
logarithm  minus  the  second  is  1.4771, 
which  is  the  logarithm  of  30.  Similarly 
for  (b)  and  (c). 


(d) 
(e) 
(f) 

.2553 

3.2553 

.7324 

(g)  1.7324 
(h)  .0128 

(c) 
(d) 

44.4 
.444 

(e)  .00444 

(f)  1.05 

(b) 

.6397 

(c)  2.4383 

(b) 

.24225 

(c)  4933 

l82  PROBLEMS  AND  STUDIES 

Problem   (22) 

(a)  log.   1.02   is   .0086;   multiplied  by  28   gives 

.2408,  which  logarithm  corresponds  to 
the  number  1.741.  The  correct  result  to 
eight  decimal  places  is  given  in  Part  IV, 
being  1.74102421. 

(b)  2.99;  to  eight  decimal  places,  the  result  is 

2.99870332. 

Problem  (23) 

(a)  log.   1.03  is  .012.8,  which  multiplied  by  45 

gives  .5760.  Then  log.  (1^1.03"')  = 
zero  minus  .5760,  or  1.4240.  The  num- 
ber corresponding  to  this  last  logarithm 
is  .265 ;  to  eight  places  the  result  is 
.26443862.    . 

(b)  .111,  and  to  eight  places,  .11129651. 

Problem   (24) 

log.  829  =2.9186 
plus  log.  76.3  -=1.8825 
plus  log.  .0484  =2.6848 
minus  log.  7.28  =  .8621 
minus  log.  25        =1.3979 

Net  result  =1.2259,  which  is  the  logarithm 
corresponding  to  the  number  16.8;  the 
result  by  actual  multiplication  and  divi- 
sion is  16.82105. 

Problem  (25) 

(a)  Amount,  $1.45161336 


(b)  "  $2.32299164 

(c)  "  $1.21899442 

(d)  "  $5.01645651 

(e)  "  $5.63210286 


present  worth,  $.68888867 

.     "  "       $.43047938 

"       $.82034830 

"       $.19934390 

"       $.17755358 


CHAPTER  XX 

PROBLEMS  IN  ANNUITIES  AND   IN   NOMINAL 
AND  EFFECTIVE  RATES 

§  232.    Problems  in  Annuities* 

(26)  Find  the  amounts  and  present  worths  of  an  an- 
nuity of  $1 : 

(a)  At  1.25%  for  30  periods 

(b)  At  1.70%  for  50  periods 

(c)  At  2.00%  for  10  periods 

(d)  At  2.40%  for  68  periods 

(e)  At  2.50%  for  70  periods 

In  Problem  (26),  a  to  e  inclusive,  assume  that  the  present 
worth  in  each  case  is  a  loan,  and  construct  a  schedule  show- 
ing the  gradual  repayment  of  this  loan  at  $1  per  period,  for 
a  few  periods  or  for  the  entire  time. 


§  233.     Answers  to  Problems  in  Annuities 

Problem   (26) 

(a)  Amount,  $36.129069 

(b)  "  $77.823037 

(c)  "  $10.949721 

(d)  "  $167.352355 

(e)  "  $185.284114 


present  worth,  $24.888906 
"       $33.501213 

"       $  8.982585 
"       $33.360671 

"       $32.897857 


In  connection  with  the  text  of  Chapters  IV  and  V. 


183 


l84  PROBLEMS  AND  STUDIES 

§  234.     Problems  in  Rent  of  Annuity  and  Sinking  Fund* 

(27)  What  is  the  rent  of  an  annuity  of  30  periods 
valued  at  $1,000  if  the  rate  of  interest  is  1.25%  per  period? 
In  other  words,  what  is  each  term  of  an  annuity  the  present 
worth  of  which  is  $1,000,  the  interest  earned  being  1.25% 
per  period  and  the  number  of  periods  30  ? 

(28)  Assume  the  same  present  worth  as  in  (27),  and 
find  the  rent  of  an  annuity  under  the  following  conditions : 

(a)  1.70%,  50  periods 

(b)  2.00%,  10  periods 

(c)  2.40%,  68  periods 

(d)  2.50%,  70  periods 

(29)  What  is  the  sinking  fund  to  be  reserved  at  the 
end  of  each  period  and  invested  at  1.25%,  to  amount  to 
$1,000  at  the  end  of  30  periods  ? 

(30)  Compute  the  sinking  funds  for  the  same  data  as 
in  (a),  (b),  (c),  and  (d),  in  (28)  above. 

(31)  What  amount  should  be  laid  aside  each  half-year 
to  amount  to  $100,000  at  the  end  of  50  years  at  4%  per 
annum,  interest  payable  semi-annually? 

(32)  What  amount  at  3%  ? 

(33)  A  father  wishing  to  make  a  gift  of  $10,000  to  his 
son,  now  15  years  old,  on  the  latter's  21st  birthday,  deposits 
a  certain  sum  at  a  trust  company,  on  a  4%  annual  basis,  on 
the  16th  and  each  succeeding  birthday,  including  the  21st, 
sufficient  to  amount  to  the  $10,000  when  the  last  deposit  is 
made.    Find  the  required  annual  deposit. 

(34)  Assume  that  after  the  annual  deposit  is  made  on 
the  18th  birthday,  the  trust  company  states  that  the  interest 
rate  thereafter  on  deposits  is  to  be  only  3%  annually.  Find 
the  annual  amount  which  should  be  deposited  on  the  19th, 


*  In  connection  with  the  text  of  Chapter  VII. 


ANNUITIES 


i8S 


20th,   and  21st  birthdays   in  order  to   reach  the  desired 
$10,000. 

(35)  On  July  1,  1914,  a  company  decides  to  accumu- 
late a  sinking  fund  of  $100,000  by  July  1,  1921,  assuming 
that  interest  on  the  fund  will  be  at  the  rate  of  4%  per 
annum.  It  is  expected  that  annual  contributions  to  the  fund 
of  $12,000  each  will  be  made  at  July  1,  1917,  1918,  1919, 
1920,  and  1921.  Find  the  two  equal  contributions  re- 
quired at  July  1,  1915  and  1916,  in  order  that  the  seven  con- 
tributions, with  accumulated  interest,  may  amount  to  $100,- 
000  at  July  1,  1921. 

§  235.    Answers  to  Problems  in  Rent  of  Annuity  and  Sink- 
ing Fund 

Problem  (27) 
$40.17854 

Problem  (28) 

(a)  $29.84967 

(b)  $111.32653 

(c)  $29.975416 

(d)  $30.39712 

Problem  (29) 

$27.67854 

Problem  (30) 

(a)  $12.84967 

(b)  $91.32653 

(c)  $5.975416 

(d)  $5.39712 

Problem  (31) 

$320.27 

Problem  (32) 
$437.06 


l86  PROBLEMS  AND  STUDIES 

Compare  the  answers  to  Problems  (27)  and  (29)"; 
(28-a)  and  (30-a)  ;  (28-b)  and  (30-b)  ;  (28-c)  and  (30-c)  ; 
and  (28-d)  and  (30-d),  respectively.  Note  that  the  differ- 
ences between  these  five  pairs  of  answers  are  in  proportion 
to  the  respective  five  rates  of  income. 
Problem  (33) 

$1,507.62 
Problem  (34) 

$1,571.53 
Problem  (35) 
$14,103.35 

§  236.     Problems  in  Nominal  and  Effective  Rates* 

(36)  If  the  interest  rate  is  12%  per  annum,  payable  in 
monthly  instalments,  what  is  the  effective  annual  rate? 

(37)  If  the  interest  is  12%  payable  semi-annually,  what 
is  the  effective  annual  rate? 

(38)  What  is  the  nominal  rate  per  annum  which,  if  paid 
semi-annually,  is  equivalent  to  an  effective  rate  of  .99505% 
per  quarter? 

(39)  (a)  If  the  nominal  rate  is  4%  per  annum,  payable 
semi-annually,  what  nominal  rate  per  annum,  payable 
quarterly,  will  produce  the  same  income  ? 

(b)  What  is  the  equivalent  nominal  annual  rate,  payable 
monthly  ? 

(40)  Interest  being  6%  per  annum,  payable  quarterly 
(the  effective  rate  per  annum  being  therefore  1.015^), 
which  is  the  more  valuable — an  income  of  $4,080,  payable 
at  the  end  of  the  year,  or  an  income  of  $4,000,  of  which 
$1,000  is  payable  at  the  end  of  each  quarter? 

(41)  Interest  being  worth  5%  per  annum  converted 


•  In  connection  with  the  text  of  Chapter  VIII. 


NOMINAL    AND    EFFECTIVE    RATES  187 

quarterly,  what  rate  should  be  paid  annually  as  an  equiva- 
lent? (Note  that  the  expressions  "payable  annually,"  "pay- 
able quarterly,"  etc.,  signify — through  custom — that  the  in- 
terest is  payable  at  the  end  of  the  year,  quarter,  etc.  When 
interest  is  paid  before  the  end  of  the  interest  period,  an 
element  of  discounting  enters  in.) 

(42)  (a)  Given  5%  as  the  effective  annual  rate;  de- 
scribe the  process  of  finding  the  effective  quarterly  rate 
equivalent  thereto. 

(b)  What  is  the  quarterly  rate  so  found? 

(c)  To  what  nominal  annual  rate  is  this  quarterly 

rate  equivalent  ? 

(43)  A  note  for  $1,000,  due  in  one  year,  is  discounted 
at  the  beginning  of  the  term,  the  net  proceeds  being  $940 : 

(a)  What  is  the  discount  rate? 

(b)  What  is  the  interest  rate  which  is  actually 

being  paid? 

(44)  If  the  above  note  were  for  six  months  and  the  net 
proceeds  were  $970,  what  would  be  the  nominal  annual  in- 
terest rate? 

(45)  Suppose  the  above  note  were  for  three  months  and 
the  net  proceeds  $985 ;  find  the  nominal  annual  interest  rate. 

§  237.     Answers  to  Problems  in   Nominal  and  Effective 
Rates 

Problem   (36) 
12.68% 

Problem   (37) 
12.36% 

Problem  (38) 
4% 


l88  PROBLEMS  AND  STUDIES 

Problem  (39) 

(a)  3.98% 

(b)  3.97% 

Problem  (40) 

The  latter,  by  $10.90 

Problem  (41) 
5.095% 

Problem  (42) 

(a)  Find  the  4th  root  of  1.05. 

(b)  1.2272% 

(c)  4.9088% 

Problem  (43) 

(a)  6% 

(b)  6.383% 

Problem  (44) 
6.186% 

Problem  (45) 
6.091% 

§  238.    Constant  Compounding 

In  §  93  it  was  stated  that  if  an  investment  on  a  6% 

nominal  annual  rate  were  compounded  every  millionth  of  a 

second,  or  constantly,  the  effective  annual  rate  could  never 

be  so  great  as  6.184%.    It  may  be  interesting  to  know  how 

to  ascertain  this  limit.    The  following  rule  gives  the  method : 

M^i  Rule :  Multiply  the  constant  quantity  .4342944819  +,  or 

\jjKj     .      so  much  thereof  as  is  necessary,  by  the  nominal  rate  per 

^^'^'       annum   expressed   decimally;   find   the   logarithm   of   the 

p  V^.    product ;  from  this  logarithm,  subtract  1,  and  the  remainder 

is  the  effective  annual  rate  required. 

For  example,  take  a  6%nominal  annual  rate.  .4342944819 
X  .06  =  .026057668914.     But  this   latter  number   is  the 


NOMINAL    AND    EFFECTIVE    RATES 


189 


logarithm    of    1.061837,    which,    diminished   by    1,    gives 
.061837,  which  is  the  limit  required.* 

§  239.    Finding  Nominal  Rate 

The  opposite  rule  for  finding  a  nominal  rate  which,  if 
compounded  an  infinite  number  of  times,  will  amount  to  a 
given  effective  rate  at  the  end  of  the  year,  is  as  follows : 

Rule:  Multiply  the  logarithm  of  the  effective  ratio  by 
the  constant  quantity  2.302585092994  +,  or  so  much  there- 
of as  is  necessary,  and  the  product  will  be  the  nominal  rate 
itself.t 

Example:  What  rate  compounded  continuously  will 
amount  to  an  effective  rate  of  6%  ?  Log.  1.06  =  .02530587 ; 
this  multiplied  by  2.302585  gives  .058270,  the  rate  required. 

§  240.     Approximate  Rules 

An  approximation  to  the  rate  may  also  be  obtained  by 

*  For  the  benefit  of  more  advanced  readers,  an  algebraic  demonstration  of 
the  rule  is  here  given: 

(-  ,  .08  \  n 
^  "r  "^   /      =e«<*,  when  n  becomes  infinite. 

\o^.  e.o«  =  .06  log.  e  =  .06  log.  2.7182818284  =  .06  (.4342944819) 
=  .026057668914  =  log.   1.061837.     Therefore,   e  •«•  =  1.061837. 


(>+f) 


Therefore,  (  1  +  :^    )  °^  1.061837,    when   n   becomes   infinite. 

The  quantity  e,  used  above,  is  the  base  of  the  Napierian  system  of  logarithms 
and  is  the  sum  of  the  infinite  series, 

'"■'^i^ti^ 


t  An  algebraic  demonstration  of  the  rule  is  as  follows : 

^^y^'^n)    =106,  when  n  becomes  infinite,  find  the  value  of  x,  i.e.,  the  nom- 
inal rate. 

(1   ,   x\  n    X  X 

'^  n  J    —^    »  when  n  becomes  infinite;  or  e    =1.06. 
Therefore,  x(log.  e)  =  log.  1.06 

Therefor.,  x  =  (log  1.06)(-^-)  =  (log.  1.06)  {:~ii^) 
=  (log.  1.06)   (2.302585092994). 


190 


PROBLEMS  AND  STUDIES 


subtracting  from  the  rate  half  its  square.     The  square  of 

.06  is  .0036,  one-half  of  which  is  .0018 ;  .06  —  .0018  =  .0582. 

Another  approximation  may  be  obtained  by  taking  the 

mean  between  the  effective  interest  rate 06 

and  the  corresponding  discount  rate .0566 

which,  added  together,  give 1166 

Half  of  this  is  the  approximate  nominal  rate 0583 


CHAPTER  XXI 

EQUIVALENT   RATES   OF   INTEREST— BOND 
VALUATIONS 

§  241.     Annual  and  Semi-Annual  Interest 

The  great  majority  of  investments  pay  interest  semi- 
annually. Occasionally  annual-interest  securities  are  offered, 
and  it  will  be  useful,  for  comparison  with  the  ordinary  semi- 
annual securities,  to  know  the  equivalent  rates.  The  fol- 
lowing table  shows  the  equivalents  for  the  more  common 
annual  rates,  the  decimals  being  carried  to  the  nearest  one- 
thousandth  of  one  per  cent. 


Table  of  Equivalent  Rates  of  Interest  Payable 
Annually  and  Semi-Annually 


Nominal  Rate 

Nominal  Rate 

Per  Annum, 

Per  Annum, 

Payable 

Payable 

Annually 

Semi-annually 

2.50% 

equivalent  to 

2.485% 

2.55% 

2.534% 

2.60% 

2.583% 

2.65% 

2.633% 

2.70% 

2.682% 

2.75% 

2.731% 

2.80% 

2.781% 

2.85% 

2.830% 

2.90% 

2.879% 

2.95% 

2.929% 

191 


192 


PROBLEMS  AND  STUDIES 


Nominal  Rate 

Nominal  Rate 

Per  Annum, 

Per  Annum, 

Payable 

Payable 

Annually 

Semi-annually 

(Continued) 

(Continued) 

3.00% 

equivalent   1 

:o     2.978% 

3.05% 

tt           t 

'       3.027% 

3.10% 

t(           ( 

'       3.076% 

3.15% 

It           ( 

'       3.126% 

3.20% 

((           i 

'      3.174% 

3.25% 

it           i 

'       3.224% 

3.30% 

it           I 

'       3.273% 

3.35% 

it           i 

'       3.322% 

3.40% 

tt           i 

'       3.372% 

3.45% 

tt           < 

*       3.421% 

3.50% 

ti           i 

*       3.470% 

3.55% 

tt           t 

'       3.519% 

3.60% 

tt           t 

'       3.568% 

3.65% 

it           I 

'      3.617% 

3.70% 

tt           i 

'       3.666% 

3.75% 

tt           t 

*       3.715% 

3.80% 

tt           ( 

*       3.765% 

3.85% 

tt           t 

'       3.814% 

3.90% 

it           t 

'       3.863% 

3.95% 

tt           t 

'       3.912% 

4.00% 

it           t 

*       3.961% 

4.05% 

it           t 

'       4.010% 

4.10% 

a                i 

'       4.059% 

4.15% 

(t               t 

'       4.108% 

4.20% 

it               t 

'       4.157% 

4.25% 

a               i 

'       4.206% 

4.30% 

it               t 

'       4.255% 

4.35% 

it               t 

'       4.304% 

EQUIVALENT  RATES  OF  INTEREST 


193 


Nominal  Rate 

Nominal  Rate 

Per  Annum, 

Per  Annum, 

Payable 

Payable 

Annually 

Semi-annually 

(Continued) 

(Continued) 

4.40% 

equivalent  to 

4:MS% 

4.45% 

((           (( 

4.402% 

4.50% 

t(           it 

4.450% 

4.55% 

tt           tt 

4.500% 

4.60% 

li           t( 

4.548% 

4.65% 

i(           it 

4.597% 

4.70% 

it           (( 

4.646% 

4.75% 

((           (( 

4.695% 

4.80% 

((           tt 

4.744% 

4.85% 

tt           tt 

4.793% 

4.90% 

tt           tt 

4.841% 

4.95% 

tt           tt 

4.890% 

5.00% 

tt           it 

4.939% 

5.25% 

it           tj. 

5.183% 

5.50% 

it          '« 

5.426% 

'5.75% 

se               a 

5.670% 

6.00% 

if                it 

5.913% 

6.25% 

a                tt 

6.155% 

6.50% 

tt                it 

6.398% 

6.75% 

it                it 

6.640% 

7.00% 

it               tt 

•6.882% 

As  an  illustration  of  the  use  of  the^above  table,  'take  the 
annual  rate  2.50%.  In  this  case,  the  square  of  1.012425, 
which  is  the  semi-annual  effective  ratio,  equals  approxi- 
mately 1.025,  the  annual  ratio  of  increase.  In  the  case  of 
the  annual  rate  4.45%,  the  square  of  1.02201  equals  ap- 
proximately 1.0445,  etc. 


194 


PROBLEMS  AND  STUDIES 


§  242.     Semi-Annual  and  Quarterly  Interest 

Quarterly  bonds  also  occur,  but  with  less  frequency  than 
semi-annual  bonds.  Some  companies,  in  order  to  induce 
holders  of  bonds  to  register  them,  pay  interest  quarterly 
after  registration,  but  semi-annually  while  in  coupon  form. 
Sometimes,  therefore,  it  is  desirable  to  know  approximately 
how  much  improvement  in  income  will  result  from  the 
quarterly  payments. 


Table  of  Equivalent  Rates  of  Interest  Payable 
Semi-Annually  and  Quarterly 


Nominal  Rate 

Nominal  Rate 

Per  Annum, 

Per  Annum, 

Payable 

Payable 

Quarterly 

Semi-annually 

2.60%      equivalent   to 

2.508% 

2.55% 

2.558% 

2.60% 

2.608% 

2.65% 

2.659% 

2.70%' 

2.709% 

2.75% 

2.759% 

2.80% 

2.810% 

2.85%' 

2.860% 

2.90% 

2.910% 

2.95% 

2.961% 

3.00% 

3.011% 

3.05% 

3.062% 

3.10% 

3.112% 

3.15% 

3.162% 

3.20% 

3.213% 

3.25% 

3.263% 

3.30% 

3.314% 

3.35% 

3.364% 

3.40% 

3.414% 

EQUIVALENT  RATES  OF  INTEREST 


195 


Nominal  Rate 

Nominal  Rate 

Per  Annum, 

Per  Annum, 

Payable 

Payable 

Quarterly 

Semi-annually 

(Continued) 

(Continued) 

3.45%      equivalent   to 

3.465% 

3.50% 

3.515% 

3.55% 

3.566% 

3.60% 

3.616% 

3.65% 

3.667% 

3.70% 

3.717% 

3.75% 

3.768% 

3.80% 

3.818% 

3.85% 

3.869% 

3.90% 

3.919% 

3.95% 

3.970% 

4.00% 

4.020% 

4.05% 

4.071% 

4.10% 

4.121% 

4.15% 

4.172% 

4.20% 

4.222% 

4.25% 

4.273% 

4.30% 

4.323% 

4.35% 

4.374% 

4.40% 

4.424% 

4.45% 

4.475% 

4.50% 

4.525% 

4.55% 

4.576% 

4.60% 

4.626% 

4.65% 

4.677% 

4.70% 

4.728% 

4.75% 

4.778% 

4.80% 

4.829% 

4.85% 

4.879% 

196 


PROBLEMS  AND  STUDIES 


Nominal  Rate 

Per  Annum, 

Payable 

Quarterly 

(Continued) 

Nominal  Rate 

Per  Annum, 

Payable 

Semi-annually 
(Continued) 

4.90% 
4.95% 

equivalent  to      4.930% 
"       4.981% 

5.00% 

t(           ( 

'       5.031% 

5.25% 

(t           I 

'       5.284% 

5.50% 

t(           ( 

'       5.538% 

5.75% 

((           i 

'       5.791% 

6.00% 

i(           i 

'       6.045% 

6.25% 

t(           ( 

6.299% 

6.50% 

it           i 

'       6.553% 

6.75% 

(t           ( 

'       6.807% 

7.00% 

((           i 

'       7.061% 

In  illustration  of  the  above  table,  take  the  rate  4.20% 
given  in  the  first  column.  The  quarterly  ratio  is  then  1.0105. 
The  square  of  this  is  1.02111025,  w^hich  is  the  semi-annual 
equivalent  earning  ratio ;  the  equivalent  semi-annual  rate  is 
2.111025%,  and  the  nominal  annual  rate  equivalent  to  the 
last-named  figure  is  approximately  4.222%. 


§  243.     Problems  in  Valuation  of  Bonds* 

In  the  following  problems,  all  bonds  are  supposed  to  be 
semi-annual,  unless  otherwise  stated. 

(46)  What  is  the  difference  between  the  cash  and  income 
rates  of: 


In  connection  with  the  text  of  Chapter  X. 


BOND    VALUATIONS  I97 

(a)  4%   bond  netting  21/2% 

(b)  3%  bond  netting  21/2^0 

(c)  5%   bond  netting  3.40% 

(d)  3%   bond  netting  3.40% 

(e)  Y%   bond  netting  4% 

(f)  5%  bond  netting  4.80% 

(g)  3.65%  bond  netting  5% 

(47)  Remembering  that  the  premium  or  discount  on  a 
bond  is  the  present  worth  of  an  annuity  of  the  difference  in 
rates,  at  the  income  rate,  and  that  problems  have  already- 
been  given  involving  the  computation  of  present  worths  at 
the  foregoing  income  rates  (Problem  26),  find  the  premium 
or  discount  on  the  following  bonds,  and  hence  their  value, 
par  being  $1,000  in  each  case : 

(a)  4%  bond  netting  2%%,  15  years 

(b)  3%  bond  netting  21/2%,  15  years 

(c)  6%  bond  netting  3.40%,  25  years 

(d)  3%  bond  netting  3.40%,  25  years 

(e)  7%  bond  netting  4%,  10  years 

(f)  5%  bond  netting  4.80%,  34  years 

(g)  3.65%  bond  netting  5%,  35  years 

§  244.     Successive  Method  of  Bond  Valuation — Problems 

By  adding  the  net  income  for  one  period  to  each  of  the 
computed  values,  and  subtracting  the  cash  interest,  find  the 
next  periodic  value  at  141/2,  241/2,  91/2,  331/2,  and  34% 
years,  respectively.  Continue  this  operation  as  many  times 
as  you  please,  and  at  any  point  you  may  prove  your  work  by 
a  fresh  computation  of  the  annuity. 

(48)  Find  the  value  of  a  4%%  bond  having  a  par  of 
$10,000,  netting  3%%,  and  having  three  years  to  run. 
From  this  initial  value,  work  out  the  values  successively 
down  to  par  at  maturity,  and  construct  a  schedule  as  in 
§122. 


198  PROBLEMS  AND  STUDIES 

(49)  Perform  the  same  operation  with: 

(a)  a  470  bond 

(b)  a  3%  bond 

(c)  a  2%  bond 

(50)  By  the  use  of  logarithms,  find  the  values  of  the 
following  bonds  of  $1,000  each  : 

(a)  4%  bond,  netting  4.50%,  95  years 

(b)  3%%  bond,  netting  3%,  401/2  years 

(c)  7%  bond,  netting  4%%,  45  years 

(d)  5%  bond,  netting  4%,  28  years 

(e)  31/^%  bond,  netting  3.80%,  100  years 

§  245.    Answers  to  Bond  Valuation  Problems 

Problem  (46) 


(a)  .75% 

(c)  .80% 

(e)   1.5% 

(b)  .25% 

(d)  .20% 

(f)  .1% 

(g)  .675% 

Problem  (47) 

(a)   $1,186.67 

(c)   $1,268.01 

(e)   $1,245.27 

(b)   $1,062.22 

(d)  $933.00 

(f)  $1,033.36 

(g)  $777.94 

Problem   (48) 

$10,282.45 

Problem   (49) 

(a)   $10,141.22 

(b)  $9,858.78 

(c)  $9,576.33 

Problem  (50) 

(a)  $890.51 

(c)   $1,480.56 

(e)  $922.88 

(b)  $1,116.77       (d)  $1,167.52 

§  246.     Bond  Valuations  by  the  Use  of  Logarithms 

The  following  will  illustrate  the  method  of  solution  by 
logarithms,  taking  (for  example)  Problem  (50-a).  Here 
the  number  of  periods  is  190,  the  difference  between  the 


BOND    VALUATIONS  Iqq 

cash  and  income  rates  per  period  is  $2.50,  and  the  income 
rate  is  2.25%  per  period.  We  must  therefore  find  the 
present  worth  (P)  of  an  annuity  of  $2.50  for  190  periods 
at  2.25%,  and  subtract  this  from  the  par  of  the  bond 
($1,000),  since  this  bond  is  at  a  discount,  the  income  rate 
being  larger  than  the  cash  rate.  The  formula  for  the 
value   of  the   discount   on   a   bond,    as  given   in   §  159, 

is(t-g)n~  (l-^i)A 

which  becomes  (2.50)(  "^  ""  1.0225^^M 
^       .0225        / 

Now,  log.  1.0225  =  .00966331668. 

Therefore,  log.  1.0225''°  =  190  X  .00966331668  = 
1.8360301692. 

Hence,  ^og.\z^^^^^]-=\og.  1-log.  1.0225"« 

=  zero  —  1.8360301692  =  2.1639698308. 

The  number  corresponding  to  this  logarithm  is  .014587128. 

The  value  of  the  discount  thus  becomes : 


2.50 


/l  --.01458Y128\ 
\  .0225  / 


which  equals  $109.49.    This  discount  when  deducted  from 
the  par  of  $1,000  gives  the  value  of  the  ix)nd,  $890.51. 

The  solution  by  logarithms  involves  considerable  "figur- 
ing,'* but  is  nevertheless  far  superior  to  any  solution  by 
ordinary  arithmetic.  The  labor  of  finding  the  present  worth 
of  an  annuity  for  190  periods  by  arithmetic  would  be 
intolerable. 


200  PROBLEMS  AND  STUDIES 

§  247.     Finding  Initial  Book  Values 

The  methods  of  finding  the  initial  book  values  of  the 
bonds  in  Schedules  (A)  and  (B)  (§  122)  are  not  shown 
in  the  text.  The  operation  is  here  given  without  logarithms, 
and  with  some  variations  in  method. 

Take  the  case  of  the  bond  in  Schedule  (A),  a  5%  bond 
for  $100,000  to  net  4%,  due  in  5  years.  The  problem  is 
to  find  the  present  value  of  an  annuity  of  $500  for  10 
periods  at  the  ratio  1.02 ;  but  in  the  present  method  we  also 
require  the  separate  present  worths  of  each  instalment  of 
$500.  These  ten  present  worths  are  the  ten  respective 
amounts  of  amortization  for  the  ten  periods  in  the  life  of 
the  bond. 

The  present  worth  of  the  first  instalment  of  $500  (i.e., 
the  first  amortization)  will  be  $500 -^  1.02^^  the  present 
worth  of  the  second  will  be  $500  -^  1.02^ ;  etc.  Since  mul- 
tiplication is  easier  than  division,  it  will  be  best  to  obtain 
first  the  value  of  1-^1.02";  500  times  this  will  give  the 
present  worth  of  the  first  instalment,  or  the  first  entry  in 
the  amortization  column.  From  the  first  amortization,  the 
second  and  following  ones  may  be  obtained  by  successive 
multiplications  by  1.02. 

To  obtain  the  value  of  1  -^  1.02'^  we  find  first  the  10th 
power  of  1.02.  After  multiplying  1.02  by  itself,  we  do  not 
again  use  it  as  a  multiplier,  but  square  the  square,  giving 
the  fourth  power.  The  4th  multiplied  by  the  4th  gives  the 
8th,  and  the  8th  multiplied  by  the  2nd  gives  the  10th  power 
of  1.02,  as  shown  on  the  following  page.  A  check  on  the 
accuracy  of  the  result  may  also  be  obtained  by  employing  the 
method  suggested  in  the  footnote  of  §  19.  In  this  latter 
case,  the  process  consists  in  finding  the  value  of  (1  +  .06)^^ 
by  the  use  of  the  algebraic  formula  known  as  the  binomial 
theorem. 


BOND    VALUATIONS 


201 


102 

102 

102 

204 

10404 
41616 
41616 

(102^)  (It  is  unnecessary  to 
repeat  the  multiplier) 

108243216 
865945728 

(102^)  =  (102^)^ 

21648643 
4329729 

(contracted  multiplication) 

324730 

21649 

1082 

649 

11716593810  1 
10404 

(102)«=(102*)^ 

11716593810 

468663752 

4686638 

12189944200    (102)^"=  (102«)  X  (102') 


§  248.    Tabular  Multiplication  and  Contracted  Division 

Next,  1  is  to  be  divided  by  1.21899442.  We  shall  use 
contracted  multiplication,  and  further  facilitate  the  work  by 
employing  the  tabular  plan.  This  consists  in  preparing  in 
advance  a  table  of  the  first  9  multiples  of  1.21899442  in  such 
a  way  that  we  are  certain  of  their  correctness.  The  use  of 
a  table  such  as  this  greatly  facilitates  accuracy  and  quickness 
in  performing  the  division  of  several  numbers  by  the  same 
divisor,  especially  in  cases  where  the  divisor  is  lengthy  and 
no  calculating  machines  are  available. 


202 


PROBLEMS  AND  STUDIES 


On  the  first  line  of  the  table  we  set  down  the  number,  and 
on  the  second  line,  its  double. 


121899442 


243798884 


Proof 


The  third  line  is  formed  by  adding  the  first  to  the  second, 
and  all  the  others  in  succession  by  adding  the  first.  The 
proof  line  is  10  times  the  original,  if  there  is  no  mistake  in 
the  work. 


121899442 


243798884 
365698326 
487697768 
609497210 
731396652 
853296094 
975195536 
097094978 


Proof 


1218994420 


The  contracted  division  consists  in  merely  subtracting 
these  multiples.  The  quotient  may  as  well  be  placed  above 
the  dividend  to  save  space. 


BOND    VALUATIONS 
Quotient     820348300 


203 


Dividend 
(8) 

1000000000 
975195536 

(2) 

24804464 
24379888|| 

(03) 

424576 
365698  1 

(4) 

58878 
48760 

(8) 

10118 
9752 

(3) 

366 
366|| 

$.8203483  is  therefore  the  present  v^orth  of  $1  due  in 
5  years ;  its  product  by  500  is  the  first  amortization : 

$410.17415 
Subtracting  this  from. ,     500. 

gives  the  compound  discount $    89.82585 

Dividing  this  by  .02  gives 4491.2925   (D-^t  =  P) 

or  the  premium,  rounded  to 4491.29 1| 

§  249.     Formation  of  Successive  Amortizations 

Our  amortization  column  v^ill  begin  v^ith  $410.17,  and 
each  successive  term  will  be  1.02  times  the  preceding,  while 
the  sum  of  the  column  must  be  $4,491.29.  To  insure  ac- 
curacy in  the  last  figure,  it  will  be  well  to  retain  at  least  the 
mills.  Having  obtained  all  the  ten  terms,  the  multiplication 
is  performed  once  more,  giving  as  a  test  $500.    The  terms 


204  PROBLEMS  AND  STUDIES 

are  again  tested  by  addition,  bringing  the  result,  $4,491.29. 
Then  the  book  values  beginning  with  $104,491.29,  and  end- 
ing with  $100,000,  are  formed  by  subtraction,  still  retain- 
ing the  mills.  In  making  up  the  schedule  the  values  are 
rounded  to  the  nearest  cent,  and  the  amortization  column  is 
made  to  correspond. 


$410,174 

$104,491,292 
410.174 

418.377 

$104,081,118 

418.377 

426.745 

$103,662,741 
426.745 

435.280 

$103,235,996 
435.280 

443.986 

$102,800,716 
443.986 

452.866 

$102,356,730 
452.866 

461.923 

$101,903,864 
461.923 

471.161 

$101,441,941 
471.161 

480.584 

$100,970,780 
480.584 

490.196* 

$100,490,196 
490.196 

$100,000,000 

Total,  $4,491,292 

•  $490,196  X  1.02  =  $500 

BOND    VALUATIONS  205 

§  250.     Test  by  Differencing 

In  a  successive  computation  like  the  one  just  given,  a 
slight  error  increases  at  every  step,  and  there  is  danger  that 
a  great  many  terms  may  have  to  be  recalculated.  The 
method  of  differencing,  applied  during  the  progress  of  the 
work,  will  form  an  efficient  check  on  all  except  the  last 
figure. 

§  251.     Successive  Columns 

To  difference  a  series,  we  first  set  down  its  terms  in  a 
first  column.  In  the  second  column  we  set  down  the  first 
differences  (Di),  of  which  the  first  line  is  the  difference  be- 
tween the  first  term  and  the  second,  the  second  line  is  the 
difference  between  the  second  and  the  third,  and  so  on. 
D2  is  composed  of  the  differences  between  these  first  differ- 
ences. D3  is  formed  from  D2  in  just  the  same  way  as  Da 
from  Di,  and  all  succeeding  differences  in  the  same  way,  to 
the  extent  required. 

The  terms  just  obtained  in  amortizing  $104,491,292 
down  to  par,  would  be  differenced  as  follows : 


Term 

D. 

D, 

D, 

410.174 

8.203 

.165 

.002 

418.377 

8.368 

.167 

.004 

426.745 

8.535 

.171 

.003 

435.280 

8.706 

.174 

.003 

443.986 

8.880 

.177 

.004 

452.866 

9.057 

.181 

.004 

461.923 

9.238 

.185 

.004 

471.161 

9.423 

.189 

.003 

480.584 

9.612 

.192 

490.196 

9.804 

500.000 

2o6  PROBLEMS  AND  STUDIES 

§  252.     Intentional  Errors 

To  demonstrate  the  utility  of  the  method,  introduce  an 
error  purposely  by  altering  one  of  the  figures  in  a  term  at 
least  three  or  four  lines  from  the  top.  Even  a  mill,  when 
all  the  differences  are  carried  out,  will  cause  violent  fluctua- 
tions in  the  column  IX  and  instantly  call  attention  to  the 
error. 

§  253.     Rejected  Decimals 

The  reason  the  fourth  column  shows  some  fluctuation 
even  though  no  errors  have  been  made,  is  that  the  last 
figure  of  a  term  is  never  accurate,  but  always  rounded 
off  or  up.  In  a  third  difference-column,  this  residue  of  error 
increases  threefold;  in  a  fourth  column,  it  may  reach  six 
times  the  original  rounding,  and,  in  the  fifth,  ten  times. 

§  254.     Limit  of  Tolerance 

The  extent  to  which  the  last  column  of  differences  may 
be  allowed  to  "waver"  will  be  learned  by  experience.  The 
next-to-the-last  column  should  be  progressive;  that  is,  it 
should  never  change  its  course  and  go  backward ;  it  should 
either  constantly  increase  or  constantly  decrease. 

It  will  be  a  useful  exercise  to  take  the  more  extended 
value,  $410.17415  (instead  of  $410,174),  multiply  it  up  to 
$500,  and  difference  the  results  out  to  5  differences.  A  very 
minute  error  will  become  enormously  magnified  and  call 
attention  to  itself. 


CHAPTER  XXII 

BROKEN   INITIAL  AND  SHORT  TERMINAL 
BONDS 

§  255.     Problems  in  Valuation* 

(51)  Suppose  the  value  of  a  4%  bond  for  15  years  on 
a  2%%  basis  to  be,  as  shown  in  Problem  (4Y-a),  $1,- 
186.66680;  what  would  be  its  value  one  month  later,  the 
time  prior  to  maturity  then  being  14  years,  11  months? 
(Since  we  are  dealing  with  half-years,  this  time  must  be 
treated  as  14%  years,  5  months,  or  15  years  less  1/6  of  the 
semi-annual  amortization  period.) 

The  theoretical,  or  mathematically  correct,  value  (§  129) 
In  the  above  case  would  be  ascertained  as  follows : 

The  ratio  of  increase  is 1.0125 

Its  logarithm  is 005  395  031  887 

This  must  be  divided  by  6,  giving .000  899  171  981 

which  is  the  logarithm  of  the  6th 

root  of  1.0125,  or  (in  other  words) 

the  logarithm  of  the  effective  ratio 

for  1/6  of  a  semi-annual  period. 
The  number  corresponding  to  the  last 

logarithm  is 1.002  072  564  8 

Multiplying  the   value   at   the   begin- 
ning  of   the    15-year   period    ($1,- 

186.66680),  by  this  number,  gives 

the  flat  value  at  14  years,  11  months, 

before  maturity $1,189,126  21 

*  In  connection  with  the  text  of  Chapter  XI. 

207 


2o8  PROBLEMS  AND  STUDIES 

Although  the  above  method  is  never  used  in  actual  buy- 
ing or  selling,  yet  it  is  proper  for  estimating  results  of 
financial  operations. 

(52)  A  firm  of  brokers  offers  $50,000  of  3%  bonds,  due 
July  1,  1929,  J  &  J,  on  a  21/2%  basis.  What  should  be  the 
price  on  September  25,  1914,  flat  or  "and  interest"? 

(53)  On  July  10,  1913,  $25,000  of  5%  bonds  due 
April  1,  1938,  A  &  O,  are  bought  at  a  price  to  yield  3.40%. 

(a)  What  is  the  flat  price  ? 

(b)  What  is  the  price  "and  interest"? 

(54)  $10,000  of  3%  bonds  due  January  1,  1938,  J  &  J, 
are  purchased  to  net  3.40%.  Find  the  price  exclusive  of 
interest  and  the  price  flat,  on  May  16, 1913. 

(55)  $6,000  of  41/2%  bonds  were  issued  in  1908,  due 
April  1,  1928,  M  &  N.  Find  the  price  "and  interest,"  on 
July  1,  1914,  on  a  4.80%  basis. 

(56)  An  investor  owns  the  four  lots  of  bonds  men- 
tioned in  Problems  (52),  (53),  (54),  and  (55),  and  has 
hitherto  carried  them  on  his  books  at  par.  He  desires  to 
have  them  adjusted  to  investment  value  as  of  December  31, 
1914.     What  will  be  the  investment  value: 

(a)  Of  each  lot? 

(b)  Of  the  aggregate? 

(57)  Find  the  amount  of  amortization  for  the  semi- 
annual period  ending  June  30, 1915  : 

(a)  On  each  of  these  lots  of  bonds. 

(b)  On  the  aggregate  of  the  four  lots. 

(58)  Taking  the  bonds  in  Problem  (53),  ascertain  their 
values  at  April  1  and  October  1,  1937,  and  thence  at  July 
1,  1937.  From  this  last  value,  (a)  amortize  to  January  1, 
1938,  (b)  and  then  for  the  broken  period  to  April  1,  1938, 
when  they  should  reduce  to  par. 

(59)  Taking  the  bonds  in  Schedule  (H)   (§141),  re- 


SHORT  TERMINAL  BONDS  209 

construct  the  schedule  so  that  the  next  date  after  May  1, 
1914,  is  July  1,  1914;  then  January  1,  1915,  and  so  on  at 
balancing  periods,,  giving  a  J  &  J  schedule  instead  of  an 
M  &  N  schedule. 

(60)  A  certain  issue  of  $100,000  of  4%  bonds  is  dated 
September  1,  1913,  and  interest  begins  at  that  date;  but  in- 
terest is  payable  on  February  1  and  August  1,  and  the  prin- 
cipal (with  4  months'  interest)  is  payable  December  1,  1917. 

(a)  What  is  the  value  of  the  bonds  on  a  3.60% 

basis  at  the  date  of  issue  ? 

(b)  What  is  their  value  on  the  same  basis  if  pur- 

chased at  December  1,  1913  ? 

(c)  At  August  1,  1917? 

(In  this  question,  note  that  the  period  at  the  beginning 
is  for  5  months,  and  not  the  usual  6  months. ) 

(61)  Make  a  schedule  running  from  December  1,  1913, 
to  maturity,  of  the  above  bonds  at  the  F  &  A  dates. 

(62)  Make  a  schedule  as  above,  but  with  J  &  J  dates, 
for  balancing  purposes. 

(63)  A  broker  offers  the  above  bonds  on  December  1, 
1913,  at  101.50  (meaning  $101.50  for  each  $100  of  par, 
which  is  the  customary  phrase),  which  he  says  will  pay 
about  3.60%.  Eliminate  any  residue  by  the  methods  in 
§§  136  to  139,  inclusive,  making  a  J  &  J  schedule*  running 
to  maturity. 

As  will  be  noted,  this  last  example  contains  all  of  the 
following  peculiarities:  short  initial  period,  odd  purchase 
date  in  that  period,  short  terminal  period,  interpolated 
balance  dates,  and  residue  to  be  eliminated. 

§  256.    Answers  to  Valuation  Problems 

Problem  (51) 

$1,189.13902  (by  the  customary  method). 


2IO  PROBLEMS  AND  STUDIES 

Problem  (52) 

$53,420.93  flat,  or  $53,070.93  and  interest. 
Problem  (53) 

(a)  $31,996.64  flat. 

(b)  $31,652.89  and  interest. 

Problem   (54) 

$9,336.43  and  interest,  or  $9,448.93  flat. 
Problem  (55) 

$5,820.34  and  interest. 
Problem  (56) 

(a)  $53,025.00;  $31,392.26;  $9,365.30;  $5,825.03. 

(b)  $99,607.59. 

Problem  (57) 

Amortization,  $87.19  and  $91.33;  accumulation, 
$9.21  and  $4.80;  net  amortization,  $164.51. 

Problem   (58) 

Value  at  January  1,  1938,  $25,098.33;  for  the 
broken  period  from  January  1  to  April  1,  ■ 
1938,  interest  on  premium  is  $1.67,  interest 
on  par  is  $212.50,  and  cash  interest  is  $312.50, 
thus  reducing  the  bond  to  par. 

Problem   (59) 

July  1, 1914,  $104,693.02 ;  January  1, 1915,  $104,- 
286.88 ;  etc. ;  July  1,  1919,  $100,245.90. 

Problem  (60) 

(a)  September  1,  1913,  $101,563.90. 

(b)  December  1,  1913,  $101,477.98. 

(c)  August  1, 1917,  $100,131.75. 

Problem  (61) 

Value  at  February  1,  1914,  $101,420.69;  etc. 


SHORT  TERMINAL  BONDS  2II 

Problem  (62) 

Value  at  January  1,  1914,  $101,449.33 ;  at  July  1, 
1914,  $101,275.34;  etc. 

Problem  (63) 

The  residue  is  $22.02,  being  the  difference  between 
$101,500.00  and  $101,477.98.  The  co- 
efficient for  elimination  of  the  residue  is 
1.0148987,  meaning  that  for  every  dollar  of 
amortization  on  the  bonds  bought  at  the  exact 
3.60%  basis,  there  should  be  added  1.48987c. 
if  the  bonds  are  bought  on  the  approximate 
3.60%  basis,  i.e.,  $101,500.00. 


CHAPTER  XXIII 

THE  USE  OF  TABLES  IN  DETERMINING  THE 
ACCURATE  INCOME   RATE 

§  257.     Bond  Tables  as  Annuity  Tables 

The  "Extended  Bond  Tables"*  can  be  used  as  an  annuity 
table  in  case  of  need,  when  the  latter  is  not  at  hand  or  when 
the  figures  in  it  are  not  sufficiently  extended  or  the  rates  not 
sufficiently  close. 

In  using  the  "Extended  Bond  Tables"  for  this  purpose, 
it  must  be  remembered  that  its  results  are  based  on  semi- 
annual payments  of  interest,  the  periods  being  half-years.  In 
the  foregoing  problems  on  annuities  where  periods  and  rates 
per  period  are  used,  in  order  to  make  use  of  the  bond  tables 
these  "periods  and  rates  per  period"  must  be  transformed 
into  years  and  rates  per  annum,  payable  semi-annually.  In 
this  manner  the  data  given  in  Problem  (26)  will  be  changed 
as  follows : 

1.25%,  30  periods,  becomes  2.50%,  15  years. 
1.70%,  50  periods,  becomes  3.40%,  25  years. 
2.00%,  10  periods,  becomes  4.00%,  5  years. 
2.40%,  68  periods,  becomes  4.80%,  34  years. 
2.50%,  70  periods,  becomes  5.00%,  35  years. 

§  258.     Premium  and  Discount  as  a  Present  Worth 

As  explained  in  Chapter  X,  the  premium  or  discount  on 
a  bond  is  nothing  more  or  less  than  the  present  worth,  at 


Spraguc's  "Extended  Bond  Tables." 
212 


DETERMINING  ACCURATE  INCOME  RATE  213 

the  income  rate,  of  an  annuity  for  the  life  of  the  bond  equal 
to  the  difference  between  the  cash  and  income  rates.  Tak- 
ing, as  an  illustration,  the  second  case  mentioned  above, 
3.40%  for  25  years,  we  turn  to  the  5%  bond  table,  page  88,* 

and  find  the  value  of  such  a  bond  to  be $1,268,009.70 

The  value  of  a  similar  bond  in  the  4%  table, 

page  54*  is 1,100,503.64 

Difference $167,506.06 

The  first  amount  results  from  a  cash  rate  of  5%  and 
an  income  rate  of  3.40%  ;  in  the  case  of  the  second  amount, 
the  cash  rate  is  4%,  with  the  same  income  rate.  The  differ- 
ence between  these  two  amounts  arises  therefore  on  account 
of  the  difference  in  cash  rates,  which,  for  a  bond  of  $1,- 
000,000,  is  $10,000  annually.  In  other  words,  the  differ- 
ence is  the  present  worth  of  an  annuity  of  $10,000  per 
annum,  payable  semi-annually,  at  3.40%  for  25  years.  Ex- 
pressed in  periods,  it  is  the  present  worth  of  an  annuity  of 
$5,000  per  period,  for  50  periods,  at  1.70%  per  period.  The 
present  worth  of  an  annuity  of  $1  per  period,  under  like 
conditions,  would  therefore  be  1/5000  of  $167,506.06,  or 
$33.501212. 

§  259.     Present  Worth  by  Differences 

Instead  of  using  the  coupon  rates  4%  and  5%,  we  might 
have  selected  3%  and  4%,  31/2%  and  41/2%,  5%  and  6%, 
or  any  other  two  rates  differing  by  1%.    For  example : 

Value  of  4%  bond,  yielding  3.40% $1,100,503.64 

Value  of  3%  bond,  yielding  3.40% 932,997.57 

Difference,  being  value  of  annuity. .........    $167,506.07 


*Sprag:ue'8  "Extended   Bond  Tables. 


214  PROBLEMS  AND  STUDIES 

There  is  a  discrepancy  of  one  cent  in  comparison  with 
the  previous  difference,  owing  to  the  rounded  decimals. 
The  reason  for  the  process  may  be  explained  as  follows : 

On  a  6%   bond  of  $1,000,000  yielding 

3.40%  the  cash,  or  coupon,  interest  is.  .$50,000 
the  net  income  is 34,000 

Difference $16,000 

In  the  case  of  a  4%  bond,  the  interest  at 

the  cash  or  coupon  rate  is $40,000 

the  net  income  is. 34,000 

Difference $6,000 

The  term  "net  income,''  as  here  used,  has  a  slightly 
different  meaning  from  its  use  in  the  schedules  in  Chapters 
X  and  XI ;  in  the  latter  case,  the  income  rate  was  applied  to 
the  book  value,  while  in  the  present  instance  it  is  applied  to 
the  par  value. 

Hence,  from  the  bond  tables  we  may  derive  the  present 
worths  of  two  annuities  of  $16,000  and  $6,000  (being  re- 
spectively $268,009.70  and  $100,503.64),  and  their  differ- 
ence must  always  be  the  present  worth  of  an  annuity  of 
$10,000.    From  the  foregoing,  we  may  state  the  following : 

Rule :  The  present  worth  of  an  annuity  of  $10,000,  pay- 
able semi-annually,  at  a  certain  income  rate,  is  equal  to  the 
difference  between  the  values  of  a  4%  and  a  5%  bond  for 
$1,000,000  at  the  same  income  rate. 

If  it  should  happen  that  the  rent  of  the  desired  annuity 
were  $5,000  instead  of  $10,000,  the  present  worth  thereof 
might  be  obtained  at  once  from  the  difference  in  values  be- 
tween 3%  and  31/2%  bonds,  or  between  31/2%  and  4% 
bonds.  Similarly,  the  difference  between  31/^  %  and  5% 
bonds  would  give  the  present  worth  of  an  annuity  of  $16,- 


DETERMINING  ACCURATE  INCOME  RATE 


215 


000;  3%  and  5%,  $20,000;  31/2%  and  6%,  $25,000;  3% 
and  6%,  $30,000;  31/2%  and  7%,  $35,000;  and  3%  and 
7%,  $40,000.  These  results  would  be  a  trifle  more  accurate 
in  the  last  figure  than  those  obtained  by  multiplying  the 
present  worths  of  the  $10,000  annuities,  since  the  multiplica- 
tion of  figures  which  have  been  rounded  increases  the  error. 

§  260.    Present  Worth  by  Division 

The  present  worth  of  an  annuity  may  also  be  obtained 
by  division  from  a  single  bond  value,  instead  of  taking  the 
difference  between  two.  We  saw  that  the  premium  on  a  4% 
bond  to  net  3.40%  is  the  present  worth  of  an  annuity  of 
$6,000,  payable  semi-annually;  therefore,  if  the  premium  be 
divided  by  6,  it  will  give  the  present  worth  of  an  annuity 
of  $1,000,  payable  semi-annually  : 

$100,503.64  -^-  6  =  $16,750.61 

§  261.    Compound  Discount  and  Present  Value  of  a  Single 
Sum 

From  the  present  worth  of  an  annuity  of  $10,000  ob- 
tained as  above,  the  compound  discount  and  the  present 
value  of  a  single  sum  for  the  same  time  and  rate  can  also 
be  ascertained.  Multiplying  the  present  worth  of  the  an- 
nuity by  the  number  of  units  in  the  rate  per  cent  gives  the 
compound  discount  on  a  single  sum  of  $1,000,000. 

$167,506.06  times  3.4  =  $569,520.60  compound  discount 
Subtract  this  from 1,000,000.00 


and  we  have $430,479.40,  which  is  the  present 

worth  of  $1,000,000  payable  in  a  single  sum  in  25  years  at 
3.40%  compounded  semi-annually.  These  computations  are 
merely  applications  of  the  two  formulas,  PX**  =  D  (§67) 
and  p  =  l  —  D  (§35).    The  last  figure  in  the  above  present 


2l6 


PROBLEMS  AND  STUDIES 


worth  is  unreliable;  as  a  matter  of  fact,  the  cents  should 
be  38. 

If  necessary,  in  the  absence  of  compound  interest  tables 
or  logarithms,  the  amount  of  a  single  sum  at  compound  in- 
terest may  be  obtained  through  the  application  of  the 
formula  a^l^p  (§35).  The  present  value  of  $1  (or  p) 
is  $.4304794;  therefore,  divide  1  by  .4304794,  using  con- 
tracted multiplication. 

4304794  )  1.0000000  (  2.3229917 

8609588 


1390412 

1291438 1     (The  sign  |    indicates  con- 

traction  or  rounding.) 

98974 

86096| 

12878 

8610 

4268 

3874 

(For  explanation  of 

contracted  multiplica- 

394 

tion,  see  §  228.) 

387  1 

7 

4]| 

3 


§  262.     Use  of  Bond  Tables  in  Compound  Interest  Problems 

The  amount  of  $1  for  50  periods  at  1.70%  per  period,  as 
above  computed,  is  $2.3229917,  and  the  compound  interest 
is  $1.3229917.  If  the  latter  amount  be  divided  by  .017,  the 
rate  of  income  for  a  single  period,  the  result  ($77.82306) 
will  be  the  amount  of  an  annuity  of  $1  for  50  periods  at 


DETERMINING  ACCURATE  INCOME  RATE 


217 


1.70%  ;  being  an  application  of  the  rule  (§  60)  A  =  1  -^i. 
Again,  when  this  result  ($77.82306,  or  A)  is  divided  by 
2.3229917  (or  a)  the  quotient  is  $33.5012  +  (or  P),  which 
is  the  present  worth  of  an  annuity  of  $1  for  50  periods  at 
1.70%.  This  is  an  application  of  the  formula  (§67) 
P  =  A  -^  a.  The  quotient  last  obtained  checks  very  closely 
with  the  result  previously  found  for  the  value  of  P, 
$33.501212.  Thus,  all  of  the  problems  in  compound  in- 
terest are  soluble  through  the  bond  tables. 

§263.     Determination  of  the  Accurate  Income  Rate* 

As  stated  (§  136),  values  of  bonds  for  each  one- 
hundredth  of  one  per  cent  of  gradation  in  the  ordinary  in- 
come rates  may  be  obtained  from  Sprague's  "Extended 
Bond  Tables."  If,  however,  an  even  more  minute  degree  of 
accuracy  is  desired  in  the  income  rate,  as,  for  example,  a 
rate  like  4.2678%,  these  tables  are  not  sufficient.  In  order 
to  develop  a  method  to  accomplish  this  result,  we  will  first 
state  the  problem  in  symbolic  form : 

Given  a  bond  on  which  there  is  a  premium  or  discount 
Q,  cash  rate  c,  and  number  of  periods  n,  what  is  the  income 
rate  i? 

Every  premium  or  discount  is  the  present  worth,  at  the 
income  rate,  of  an  annuity  of  n  terms,  each  instalment  of 
which  is  the  difference  between  the  cash  and  income  rates; 
in  other  words,  it  is  the  present  worth  of  an  annuity  of  $1 
multiplied  by  the  difference  in  rates  (§118).  Writing  P 
for  the  present  worth  of  an  annuity  of  $1,  we  have  the 
equation :  Q  =  P  X  (c — i) .  The  terms  c  and  i,  in  the  great 
majority  of  bonds,  theoretically  refer  to  the  rates  for  semi- 
annual periods.  In  practice,  however,  a  4%  rate  or  a  5% 
rate  means  an  annual  nominal  rate,  irrespective  of  the  fact 
that  the  coupons  are  semi-annual.    In  order  to  conform  to 


Compare  text  of  §§  135,  136. 


2i8  PROBLEMS  AND  STUDIES 

commercial  usage,  we  will  alter  the  equation  by  halving  the 
P  and  doubling  the  (c — i)  ;  the  equation  then  becomes : 
Q  =  1/2?  X  {2c— 2i).  With  this  change,  the  value  of  the 
right-hand  member  is  not  altered,  and  there  is  the  advantage 
that  the  quantities  2c  and  2i  represent,  respectively,  the 
nominal  annual  cash  and  income  rates. 

§  264.    Assumed  Trial  Rate 

In  the  equation  given  above,  the  premium  or  discount  Q 
is  known,  and  the  cash  rate  c  is  also  known.  There  is  there- 
fore, in  reality,  but  one  unknown  quantity,  the  income  rate 
i,  since  P  can  be  ascertained  when  once  the  value  of  i  is 
known.  It  is  evident  that  if  we  divide  Q  by  %P  (which 
latter  we  will  hereafter  call  the  trial  divisor),  we  shall  find 
the  difference  in  rates.  Let  us  assume  the  rate  of  income 
to  be  any  rate  whatever,  and  then  calculate  the  trial  divisor 
at  that  rate.  Then,  since  the  product  of  %P  times  (2c — 2i) 
is  the  constant,  or  known,  quantity  Q,  we  have  the  following 
chain  of  reasoning:  If  the  assumed  income  rate  is  too 
small,  P  will  be  too  large,  the  difference  in  rates  will  be  too 
small,  and  the  ascertained  income  rate  will  be  too  large ;  and 
vice  versa  if  the  assumed  income  rate  is  too  large.  Taking 
now  this  first  ascertained  rate  as  the  new  assumed  rate,  we 
may  find  a  second  ascertained  rate,  and  so  on,  as  many  times 
as  we  please,  the  proceeding  being  something  like  the  swing- 
ing of  a  clock  pendulum,  except  that  each  swing  is  shorter 
than  the  preceding  one,  since  the  successive  ascertained  rates, 
one  after  another,  more  nearly  approach  the  true  income 
rate.  We  may  slightly  modify  any  rate  in  order  to  make 
the  work  easier;  if  we  are  fortunate  in  selecting  our  first 
trial  rate  near  the  true  rate,  fewer  successive  approximations 
will  be  necessary. 

For  the  purpose  of  computing  the  value  of  the  tria! 
divisor   (%P),  a  table  of  bond  values  may  be  used  for 


DETERMINING  ACCURATE  INCOME  RATE  219 

the  first  two  or  three  approximations,  by  taking  the  differ- 
ence between  the  values  (at  the  same  income  rate)  of  a 
3%  bond  and  a  4%  bond,  or  of  some  other  pair  of  bonds 
whose  nominal  annual  cash  rates  differ  by  1%. 

§265.     Application  of  Assumed  Trial  Rate — Bond  Above 
Par 

As  an  example,  we  will  take  a  6%  semi-annual  bond  for 
$100,  due  in  50  years  and  sold  at  133,  to  find  the  income 
rate.  With  so  large  a  premium  as  33,  the  income  rate  is 
evidently  much  less  than  6%  ;  let  us  assume  4%.  From  the 
bond  tables  we  find  that  the  value  of  a  5%  bond,  due  in 
50  years,  and  earning  4%,  on  a  par  of  100,  is. . .  .$121,549 
The  value  of  a  similar  bond  earning  only  4%  is 

par,  or • 100.000 

The  difference  is  the  present  worth  of  an  annuity 
of  50c.  (the  difference  between  the  semi-annual 
cash  and  income  rates)  for  100  periods  at  2% 
per  period $21,549 

The  present  worth  of  a  similar  annuity  of  $1,  or 

P,  is $43,098 

%P,  the  first  trial  divisor,  is  therefore $21.55 

33.00^21.55  =  1.531,  the  difference  in  rates.  6% — 
1.531  =  4.469%,  the  new  trial  rate.  Taking  4.45%  as  more 
convenient,  the  new  trial  divisor  is  19.98.  33.00  -^  19.98  = 
1.651.  6%  —  1.651%  =4.349%.  For  this  new  rate  (or 
4.35%),  we  find  that  20.315  is  the  trial  divisor.  33.00-^ 
20.315  =  1.6244.  6%  —  1.6244%  =  4.3Y56%.  Next  using 
4.37%,  the  trial  divisor  is  20.25.  33.00-^20.25  =  4.37, 
almost  exactly,  so  that  the  use  of  4.37%  as  an  assumed  or 
trial  rate  leads  to  it  again  as  an  ascertained  rate;  in  other 
words,  the  rate  4.37%  reproduces  itself,  which  shows  that 


'/' 


220  PROBLEMS  AND  STUDIES 

we  have  now  found  the  correct  rate.  The  value  of  the  bond 
at  4.37%,  as  computed  by  logarithms,  is  $133.0069,  an 
error  of  less  than  one  cent. 

§  266.     Variations  in  Assumed  Rates 

The  example  in  §  265  is  an  illustration  of  what  we  have 
previously  pointed  out ;  that  is,  that  the  results  always  swing 
to  the  opposite  side  of  the  true  rate.  If  the  trial  rate  is  too 
large,  the  ascertained  rate  will  be  too  small,  and  the  true 
rate  will  lie  between  them.  The  successive  rates  were  4%, 
4.469%,  4.349%,4.3756%,  and  4.37%.  4.37%  lies  between 
any  pair  of  these  rates  except  the  last  two,  where  one  rate 
coincides  with  4.37%.  The  foregoing  is  always  the  case 
with  bonds  above  par.  With  bonds  below  par  it  is  different ; 
here  the  true  rate  is  always  larger  than  the  last  approxima- 
tion. The  ascertained  rate  may  be  carried  to  many  decimal 
places,  but  it  never  quite  overtakes  the  true  rate.  The  case 
is  somewhat  analogous  to  a  circle  having  an  inscribed 
polygon.  We  may  increase  the  number  of  sides  of  the 
polygon  indefinitely,  but  its  area  will  never  quite  equal  the 
area  of  the  circle. 

§267.    Application  of  Assumed  Trial  Rate — Bond  Below 
Par 

As  an  example  of  a  bond  below  par,  take  a  3%  bond 
payable  in  25  years.  If  purchased  at  88.25,  what  is  the 
income  rate  ?  The  following  may  be  the  steps,  the  dividend 
being  always  11.75,  the  discount: 

Trial  rates 3.70%         3.725%     3.7265% 

Trial  divisors 16.2190       16.175       16.17245 

Ascertained  rates.     3.7244%     3.7264%  3.7265% 

Since  3.7265%  reproduces  itself,  it  must  be  correct  to 
the  4th  decimal.  Tested  by  logarithms,  the  value  of  a  3% 
bond  for  25  years  yielding  3.7265%  proves  to  be  $88.25018. 


DETERMINING  ACCURATE  INCOME  RATE  221 

§  268.     Trial  Rates  from  Bond  Tables 

While  the  method  of  trial  rates  is  correct  in  theory,  it 
may  be  greatly  facilitated  in  practice  by  first  locating  by 
means  of  bond  tables*  the  required  income  rate  between  two 
rates  one-hundredth  of  one  per  cent  apart.  The  results  will 
be  so  close  that  simple  interpolation  (explained  in  Chapter 
XXXI)  will  suffice  for  at  least  seven  decimals,  and  the 
laborious  divisions  necessary  in  the  foregoing  method  will 
be  avoided. 

§  269.    Use  of  Bond  Tables 

For  example,  let  it  be  required  to  find  the  income  rate  of 
a  4%  bond  for  $1,000,000  due  in  100  years,  bought  for  $1,- 
264,806.66.  From  the  4%  table,  we  find  that  the  income 
rate  must  lie  between  3.10%  and  3.15%.  The  values  corre- 
sponding to  these  rates  are  as  follows : 

3.10% $1,276,929.04 

3.15% 1,257,990.62 

1/5  of  the  difference  being  $3,787,684,  we  may  roughly 
estimate  the  intermediate  values  as  follows : 

3.11% $1,273,141.35 

3.12% 1,269,353.66 

3.13% 1,265,565.98 

3.14%.. 1,261,778.30 

The  required  rate  must  lie  between  3.13%  and  3.14%; 
the  difference  in  rates  lies  between  .87%  and  .86%.  Correct- 
ing the  above  intermediate  values  by  the  colored  pages  in 
the  bond  tables,*  we  have : 

Premium  at  3.13% $265,505.52 

Premium  at  3.14% 261,738.09 

Premium  at  the  required  rate. .     264,806.66 

Since  any  two  premiums  at  the  same  income  rate  are 

•Sprague's  "Extended  Bond  Tables." 


222  PROBLEMS  AND  STUDIES 

directly  proportional  to  the  difference  between  the  cash  and 
income  rates,  we  have  the  following  proportions  : 

at  3.13%— $265,505.52  :  $264,806.66  :  :  .87%  :  x%  (x  =  .867/09998) 
at  3.14%  — $261,738.09  :  $264,806.66  :  :  .86%  ;  x%  (x  =  .870082484) 

At  the  same  premium  on  each  bond  ($264,806.66),  we 
see  from  the  above  two  proportions  that  the  following  facts 
prevail  with  reference  to  the  rates : 

Income  Rate  Cash  Rate 

3.13%     corresponds  with    3.997709998% 
3.14%  "  "        4.010082484% 

Our  problem  is  to  determine  the  income  rate  correspond- 
ing with  a  cash  rate  of  4%,  the  premium  still  being  the 
same.  For  this  purpose,  the  method  of  interpolation  will  be 
sufficiently  exact,  and  we  may  form  a  proportion  as  follows : 

x%   :  .01%   ::  .002290002%   :  .012372486% 

The  unknown  term  of  the  proportion  is  found  to  be 
.0018509%,  which  added  to  3.13%  gives  3.1318509%  as 
the  income  rate  corresponding  to  a  4%  cash  rate.  The  ac- 
curate value  of  the  bond  computed  to  ten  decimal  places  at 
the  income  rate  of  3.131851%  is  $1,264,806.6645. 


CHAPTER  XXIV 

DISCOUNTING* 

§  270.    Table  of  Multiples 

Discounting  may  be  performed  as  well  by  multiplication 
as  by  division,  and  multiplication  is  preferable  as  being  the 
more  direct  and  compact  process.  In  Table  VI  (§  383)  are 
the  reciprocals  of  all  usual  ratios  of  increase.  Multiplying 
by  .9803921568,  for  example,  will  give  the  same  result  up 
to  a  certain  number  of  places,  as  dividing  by  1.02.  Using 
the  tabular  plan,  we  have  this  table  : 


98 
196 
294 
392 
490 
588 
686 
784 
882 


039 
078 
117 
156 
196 
235 
274 
313 
352 


216 
431 

647 
863 
078 
294 
510 
726 
941 


We  will  take  as  an  illustration  a  5%  bond,  yielding  4%, 
both  the  coupons  and  the  income  being  on  a  semi-annual 
basis.  The  amounts  receivable  at  maturity  are  $100,000.00 
of  principal  and  $2,500.00  of  coupons,  a  total  of 
$102,500.00.  The  discounting  process  would  then  be  as 
follows : 


In  connection  with  text  of  §  143. 

223 


224  PROBLEJMS  AND  STUDIES 

102,500.00 


98,039.22 

1,960.78 

490.20 

100,490.20 
2,600.00 

102,990.20 

98,039.22 

1,960.78 

882.35 

88.24 

.20 

100,970.79 
2,500.00 

103,470.79 

98,039.22 
2,941.18  etc. 


There  is  an  error  of  1  cent  in  the  value  100,970.79 ;  this 
could  easily  have  been  prevented  by  carrying  out  into  mills. 
For  long  operations  it  is  always  advantageous  to  use  a  few 
spare  places  beyond  those  retained  in  the  final  result. 

§  271.    Present  Worths  of  Interest-DifiFerence 

Still  greater  brevity  will  be  attained  by  working  out  first 
the  items  of  amortization,  or  present  worths  of  the  difference 
between  the  cash  and  income  rates.  The  present  worths  of 
the  interest-difference  500  are  obtained  as  follows,  using 
fewer  figures  and  less  labor  than  in  the  preceding  example : 


DISCOUNTING  225 

600.000 


490.  X9 6  %  year  before  maturity 


392.157 

88.235 

.098 

.088 

6 


480.584  1  year  before  maturity 


392.157 

78.431 

.490 

.078 

4 


471.160  1%  years  before  maturity 

392.157 

68.627 

980 

98 

59 

461.921  2  years  before  maturity 

Writing  these  down  in  reverse  order,  the  amortization 
column  of  the  schedule  is  filled : 

461.92 
471.16 
480.58 
490.20 


1903.86 


226  PROBLEMS  AND  STUDIES 

The  value  at  two  years  before  maturity  is  therefore 
$101,903.86,  and  the  schedule  may  be  further  filled : 


Book 

Par 

Amortization 

Value 

Value 

$101,903.86 

$100,000.00 

$461.92 

101,441.94 

471.16 

100,970.78 

480.58 

100,490.20 

490.20 

100,000.00 

For  practice,  any  of  the  Problems  (52)  to  (55),  inclu- 
sive, may  be  worked  over  backwards. 

§  272.     Discounts  from  Tables 

If  the  rate  is  one  of  those  embraced  in  Table  II  (§  379), 
and  the  difference  of  interest  is  a  simple  number,  the  process 
is  still  easier.  Here  the  present  worths  of  500  for  various 
numbers  of  periods  at  2%  per  period  are  required.  In  Table 
II  we  find  these  present  worths  for  $1 ;  pointing  off  3  places 
to  the  rig-ht  gives  the  corresponding  values  for  $1,000,  and 
halving  this,  all  in  the  one  operation,  gives  the  successive 
figures  required : 

4  periods  .92384643X  1000  -^  2  =461.9227|| 
3        "      .94232233  "  471.1612|| 

2        "      .96116878  "  480.6844|| 

1        "      .98039216  "  490.1961|| 


1903.8644|I 


§  273.    Reussner's  Tables 

Reussner's  "True  Discount  Tables"  give  multipliers  for 
each  day,  from  1  to  180,  carried  to  8  places,  for  a  great 
number  of  usu^l  rates,  and  will  much  facilitate  discounting 


DISCOUNTING 


227 


for  fractional  periods.  In  the  example  in  the  text,  it  gives 
.99009901  opposite  90  days  at  4%,  with  the  following 
result : 

102,600.000 


99,009.901 

1,980.198 

495.050 


101,485.149,  the  same  as  in  the  text  of  §  144. 


CHAPTER  XXV 

SERIAL   BONDS 

§  274.     Problem  in  Valuation  of  Serial  Bonds 

(64)  A  city  issues  ten  4%  bonds  for  $10,000  each,  A&O, 
on  April  1,  1914,  maturing  as  follows :  $10,000  on  April  1, 
1916;  $10,000  on  April  1,  1918,  and  so  on— $10,000  each 
alternate  year,  the  last  $10,000  on  April  1,  1934.  They 
are  sold  at  108.33,  the  purchaser  believing  that  he  has  a 
3.10%  investment.    How  near  right  is  he? 

As  the  average  time  of  the  bonds  is  11  years,  it  might 
be  inferred  that  the  true  value  of  the  series  was  the  value 
of  a  single  bond  of  $100,000  due  in  1925,  which  would  be 
$108,334.54;  but  this  is  fallacious.  The  true  price,  obtained 
by  adding  together  all  the  separate  tabular  values,  is  always 
less. 

At  3.10%,  the  values  of  the  bond  at  varying  due  dates 
are  as  follows : 


Due  Dates 

Values 

1916 

$10,173.24 

1918 

10,336.13 

1920 

10,489.31 

1922 

10,633.35 

1924 

10,768.79 

1926 

10,896.16 

1928 

11,015.92 

1930 

11,128.53 

1932 

11,234.43 

1934 

11,334.01 

Total  series 

$108,009.87 

228 


SERIAL  BONDS 


229 


It  is  evident  that  the  purchaser  should  have  paid  108.01 
instead  of  108.33,  and  that  on  the  latter  price  he  will  earn 
less  than  3.10%.    How  much  less,  is  to  be  ascertained. 

The  value  at  3.10%  might  have  been  carried  out  further 
in  decimals  to  the  limit  of  the  tables,  giving  $108,009.8686. 

The  values  at  3.05%  will  next  be  copied  down. 


Due  Dates 

Values 

r   2 

years 

$10,182.9714 

4 

10,355.1945 

6 

10,517.3006 

8 

10,669.8840 

3.05% 

10 

10,813.5041 

Basis  1 

12 

10,948.6875 

14 

11,075.9297 

16 

11,195.6973 

18 

11,308.4294 

20 

11,414.5391 
$108,482.1376 

§  275.    Inter-rates 

The  inter-rates,  3.06%,  3.07%,  3.08%,  and  3.09%,  can 
now  be  obtained  in  bulk  without  determining  the  values  for 
separate  years,  according  to  the  directions  on  page  123  of 
Sprague's  ^'Extended  Bond  Tables." 

Find  the  difference  between ,.  .$108,482.1376 

and ; 108,009.8686 


which  is ., $472.2690 

ys  of  this  is 94.4538 

Subtracting  from  $108,482.1376  succes- 
sively ys,  }i,  Ys,  and  fs ,  we  have  the 
approximate  values  for  3.06%  ... $108,387.6838 

for  3.07% 108,293.2300 

But  it  is  unnecessary  to  go  further ;  it  is  evident  that  the 
effective  rate  is  a  little  below  3.07%. 


230 


PROBLEMS  AND  STUDIES 


§  276.     Table  of  Differences 

The  value  given  at  the  basis  of  3.07%  is  approximate, 
and  we  can  get  a  corrected  value  by  applying  the  rule  given 
on  page  122  of  the  tables,*  viz. :  ''To  correct  any  terminal 
2  or  7,  subtract  II/2  times  the  difference  and  then  add  1/10 
the  sub-difference."  The  following  table  is  derived  from 
pages  146  to  149,  inclusive,  of  the  bond  tables,*  and  shows 
the  differences  and  sub-differences  in  the  case  of  a  4%  bond 
of  $1,000,000  at  the  income  rates  of  3.05%  and  3.10%. 


Dates  of 

Differences  at 

Differences  at 

Sub- 

Maturity  of  Bond 

3.05%  Basis 

3.10%  Basis 

Differences 

2 

years 

$    .09 

$    .09 

4 

.33 

.33 

6 

.69 

.69 

8 

1.17 

1.16 

$.01 

10 

1.74 

1.73 

.01 

12 

2.40 

2.39 

.01 

14 

3.14 

3.12 

.02 

16 

3.95 

3.92 

.03 

18 

4.82 

4.78 

.04 

20 

Total 

5.74 

5.68 

.06 

$24.07 

$23.89 

$.18 

On  account  of  the  fact  that  each  of  the  bonds  in  question 
has  the  par  of  $10,000  and  not  $1,000,000,  the  tabular 
difference  for  the  rate  3.07%  becomes  $.2407,  and  the  sub- 
difference  $.0018;  1%  times  the  difference  equals  $.3611, 
and  1/10  of  the  sub-difference  is  $.0002.  The  corrected 
value  at  3.07%  therefore  becomes : 


Sprague's   "Extended   Bond   Tables. 


SERIAL  BONDS 

$108,293.2300  —  $.3611  +  $.0002  =  $108,292.8691 
The  residue  to  be  eliminated  is 37.1309 


231 


making  the  price  paid $108,330.0000 


§  277.     Successive  Method 

The  values  at  the  basis  of  3.07%  must  next  be  worked 
out  for  each  period  down  to  the  last  maturity. 

Value  at  April  1,  1914 $108,292.8691 

X  1.01535 1,082.9287 

541.4643 

32.4879 

5.4146 

less         2,000.0000 


Value  at  October  1,  1914,  $107,955.1646 

X  1.01535 1,079.5516 

539.7758 

32.3865 

5.3978 

less         2,000.0000 


Value  at  April  1,  1915 $107,612.2763 

X  1.01535 1,076.1228 

538.0614 

32.2837 

5.3806 

less         2,000.0000 

Value  at  October  1,  1915 . .    $107,264.1248 


At  April  1,  1916,  1918,  etc.,  at  intervals  of  two  years, 
the  book  value  will  be  further  diminished  to  the  extent  of 
the  principal  of  the  bonds  maturing  at  these  respective  dates. 


232  PROBLEMS  AND  STUDIES 

§  278.     Balancing  Period 

But  it  may  be  that  balancing-period  figures  are  wanted, 
say  J  &  J.    In  that  case,  the  value  on  July  1,  1914,  is  half- 
way between ,  $108,292.8691 

and 107,955.1646 

or   $108,124.0168 

with  which  we  continue —  1,081.2402 

540.6201 

32.4372 

5.4062 

less         2,000.0000 

Value  at  January  1, 1915 . .   $107,783.7205 

1,077.8372 

538.9186 

32.3351 

5.3892 

less         2,000.0000 

Value  at  July  1,  1915 $107,438.2006 

1,074.3820 

537.1910 

32.2315 

5.3719 

less         2,000.0000 

Value  at  January  1, 1916. .  $107,087.3770 

§  279.    First  Payment  in  Series 

We  have  now  reached  a  point  where  a  broken  terminal 
period  occurs,  as  to  the  first  $10,000  due  April  1,  1916, 
and  we  must  follow  the  directions  of  §  88,  with  tfiis  modifica- 
tion: that  the  $10,000  and  the  remaining  $97,087.38  must 
be  treated  separately,  the  reason  being  obvious. 


SERIAL  BONDS 


233 


$107,087.3770 
Amount  of  principal  due  April  1     10,000.0000 


Remainder  ; $97,087.3770 

The  usual  procedure — 

970.8738 

485.4369 

29.1262 

4.8544 

$10,000  X  .007675  (3  months)  . .  76.7500 


$98,654.4183 
Income  2%  on  $90,000 

1%  on  $10,000  = 1,900.0000 


Value  at  July  1,  1916 $96,754.4183 

This  will  exemplify  the  process  when  the  principal  of 
one  of  the  serial  bonds  is  paid  off. 


§  280.     Elimination  of  Residue 

There  is  a  residue  of  $37.1309  to  be  eliminated,  for 
which  we  shall  use  the  third  method.  A  total  premium  of 
$8,330  is  to  be  amortized,  while  the  3.07%  basis  will  amor- 
tize only  $8,292.8691.  The  proportion  is  8,330  -^  8,292.8691 
=  1.0044784.  A  table  formed  from  this  will  give  the  fol- 
lowing multiples : 

1004478 


2008957 
3013435 
4017914 
5022392 
6026870 
7031349 
8035827 
9040306 


234  PROBLEMS  AND  STUDIES 

The  amortization  at  3.07%  for  the  fractional  period  and 
the  4  full  periods  is  as  follows : 

168.8523   (April  1  to  July  1,  1914) 

340.2963 

345.5199 

350.8236 

332.9587 

and  as  adjusted  for  elimination  as  follows,  the  eliminands 
appearing  in  the  top  line  and  the  eliminates  in  the  bottom 
line: 

1688523         3402963         3455199         3508236       3329587 

1004478  3013435  3013435  3013435  3013435 

602687  401791  401791  502239  301344 

80358  2009  50224  8036  20090 

8036  904  5022  201  9040 

502  63  199  36  502 

23  87 

1696084         3418202         3470671         3523947      3344498 

As  thus  computed,  the  adjusted  amounts  of  amortiza- 
tion would  be  as  follows : 

April  1,  1914,  to  July  1,  1914 $169.61 

July    1,  1914,  to  Jan.  1,  1915 341.82 

Jan.    1,  1915,  to  July  1,  1915 347.07 

July    1,  1915,  to  Jan.  1,  1916 352.39 

Jan.    1,  1916,  to  July  1,  1916... 334.45 

Total  for  4%  years $1,545.34 

§  281.     Schedule 

The  schedule  will  then  be  made  up  as  follows  to  this 
point : 


SERIAL  BONDS 


235 


o  05 

O     tH 


as 


o 

H 
N 

H 
PS 
O 

< 

O 

W 
.-1 

W 
u 

CO 


05 


00 

tH 
U     CO 

tn    "^ 

O  .  >% 
PQ  ^  ^03 
-^    »—    .^ 


C/} 


o 
o 

o 
o 

o 

€/9- 


<      Oh 
\ — ;    CO 


:3  o 


0                 0 

0               0 

l-l 

0                   2 

a 

0                   0 

Pk 

0               0^ 

0^                0 

0                            05 

T-{ 

m 

0  Oi  t^  0  tH  CD 

0  CO  10   lO-  tH  CO 

0  B, 

0   0  00   rH   Oi  rJ^ 

CO   <^   rH  1>-  rH  00 

0    nj 

CO   rH   00   ThI   rH  Ir- 

m> 

00"  00"  ^-"  J>^  i>^  0 

000000 

tH   rH   rH   rH   rH 

€€ 

)■ 

N 

•  rH  oq  Ir-  Oi  0 

*f     C 

CD   00   0   CO   tH 

S.2 

Ci   rH*  i>^  Cq   t:H 

S  rt 

CD   TJH   ^   iO   CO 

< 

rH   CO  CO   CO   CO 

0^ 

Oi  00   CO   rH  VO 

CO  rH   Oi  CD   10 

ni 

0  06   C^*  1>^  lO 

-^ 

CO   lO  iO   -^  CD 

00  CD  CD  CD   iO 

aJ*^ 

^^  T-i  -rH   T-i  T-i 

iz; 

00000 

00000 

if 

00000 

00000 

0   0   0  0  Oi 

rH   CN   C^l  C?^  rH 

1— 1 

<(-> 

en 

0 

u 

,__, 

0 

a 

TjT          10           CD 

rH           rH           rH 

Oi           Oi           OS 

r- 

<          rH          iH           II 

236  PROBLEMS  AND  STUDIES 

The  premium  is  now  $6,784.66,  and  the  premium  at 
3.07%  is  $6,754.42,  which  we  test  as  follows: 

1.004478 
X  675442 


602687 

70313 

5022 

402 

40 


Proof  678466 

§  282.     Uneven  Loans 

The  terms  of  a  series  of  bonds  need  not  necessarily  be  of 
like  amount.  Suppose  the  payments  in  the  above  example 
were: 

$10,000  in  1916 

$20,000  in  1918 

$30,000  in  1920 

$40,000  in  1922 


$100,000 
and  it  were  desired  to  find  the  value  at  3.10% ;  the  process 
would  be : 

$10,173.2358  $10,173.2358 

10,336.1340  X  2  20,672.2680 

10,489.3124  X  3  31,467.9372 

10,633.3506  X  4  42,533.4024 


Value  of  series  $104,846.8434 


The  formation  of  the  schedule  would  be  precisely  an- 
alogous to  that  already  given. 


SERIAL  BONDS 


237 


§  283.     Tabular  Methods 

Most  serial  bonds  run  by  years,  an  equal  amount  being 
payable  annually.  Where  the  rate  is  one  ending  in  5  or  0, 
and  the  values  for  exact  interest  periods  are  required,  not 
for  intermediate  periods,  a  simpler  process  may  be  used, 
copying  values  direct  from  the  tables.  For  example,  a  series 
of  five  4%  bonds  of  the  par  value  of  $1,000  each,  J  &  J, 
issued  July  1,  1914,  payable  on  each  first  of  July,  1915  to 
1919,  is  sold  on  a  3.50%  basis. 

Set  down  in  two  columns  the  first  ten  values  from  the 
tables ;  then  add  and  subtract  successively,  as  follows : 


1/2  yr.  $1002.457 

1  yr.  $1004.872 

lYz            1007.245 

2     1009.577 

2%     1011.870 

3     1014.122 

31/2     1016.337 

4     1018.513 

4%     1020.651 

5     1022.753 

$5069.837 

Jan.  1,  1915,    $5058.560 


1020.651 


Jan.  1,  1916,  $4037.909 


1016.337 


Jan.  1,  1917,  $3021.572 


1011.870 


Jan.  1,  1918,  $2009.702 


1007.245 


1022.753 


$4047.084  July  1,  1915 


1018.513 


$3028.571  July  1,  1916 


1014.122 


$2014.449  July  1,  1917 


1009.577 
$1004.872  July  1,  1918 


Jan.  1,  1919,  $1002.457 


238 


PROBLEMS  AND  STUDIES 


§  284.     Formula  for  Serials 

The  total  value  of  an  annual  series  may  be  obtained  by 
the  following  formula : 

Let  m  be  the  number  of  different  maturities  and  n  the 
number  of  the  periods  the  last  bond  has  to  run.  Let  r,  for 
brevity,  represent  the  ratio  of  increase,  instead  of  1  +  i. 
The  powers  of  r  are  obtainable  from  Table  I  (§  378),  or  by 
logarithms.  The  principal  of  each  bond  being  $1,  the 
formula  would  read : 

^^^  — /^2_fy7n  K  ^'  X  (c  —  i) 

In  the  preceding  example  m^  6,n  =  10,  r  =  1.0175^  i  = 
.0175,  c  =  .02,  c  —  i  =  .0025. 

From  Table  I*  or  from  the  "Extended  Bond  Tables"t : 

1.03530625 

1.18944449 

1.18944449 

.18944449 

.03530625 

.18944449 


r^ 

= 

y5m  = 

^10  _ 

r^  = 

^10  = 

Therefore :  r'^  -  1 

= 

r'-l 

= 

^2m_i 

_ 

{r'-l)r'' 

= 

^2in  _  1 

=  n 

.03530625  X  1.18944449 
4.511139 

4.511139  =  .488861 


(m—  /o''^~x"^n  ^^ *'  ==  -488861  -^  .0175  =  27.93491 

Value  of  series  =  5  +  (27.93491  X  .0025)  =  5.0698373 
which  is  the  result  already  obtained  by  addition. 

This  formula  will  seldom  be  of  use  except  in  the  case  of 
a  very  complex  rate  not  comprised  in  the  tables.  It  will  then 
involve  the  computation  of  three  powers  of  r  by  logarithms. 


§  378.     t  Sprag ue's  "Extended  Bond  Tables. 


SERIAL  BONDS  239 

§  285.     Problems  in  Valuation  of  Serial  Bonds 

The  following  problems  may  be  solved  in  either  of  the 
ways  discussed : 

(65)  A  corporation  issued  a  series  of  ten  $1,000  bonds, 
5%,  M  &  N,  on  May  1,  1913,  payable  each  May  1,  1921  to 
1930.    What  is  the  value  on  a  3.60%  basis : 

(a)  On  May  1,1918? 

(b)  On  July  1,  1918? 

(c)  On  August  23,  1918? 

(66)  Find  the  values  as  above,  but  on  a  4%  basis. 

§  286.     Answers  to  Problems  in  Valuation  of  Serial  Bonds 

Problem  (65) 

(a)  $10,897.40 

(b)  $10,962.79  flat. 

(c)  $11,019.45  flat. 

Problem  (66) 

(a)  $10,630.42 

(b)  $10,701.29  flat. 

(c)  $10,762.71  flat. 


CHAPTER  XXVI 

OPTION  OF  REDEMPTION 

§  287.     Method  of  Calculating  Income  Rate* 

The  rate  of  income  on  a  bond  subject  to  a  right  to  redeem 
at  an  earlier  date  than  that  of  actual  maturity  and  on  pay- 
ment of  a  premium,  can  be  ascertained  by  means  of  tables. 
Only  the  income  which  is  certain  must  be  calculated  upon  in 
advance;  hence  there  will  always  be  a  contingent  profit 
which  may  be  realized. 

For  example,  suppose  the  bond  to  be  a  4%%  one  abso- 
lutely due  in  30  years  but  redeemable  at  105  after  20  years; 
issued  1905,  redeemable  1925,  payable  1935. 

In  order  to  determine  where  the  redemption  is  a  benefit 
and  where  it  is  a  disadvantage,  we  must  suppose  ourselves 
to  be  in  1925  at  the  redemption  date.  This  bond  now  has 
10  years  to  run.  Turning  to  the  4%%  bond  table,t  under 
10  years,  we  find  that  1.05  is  the  price  almost  exactly  at  a 
3.89%  basis.  Therefore,  if  the  bond  is  bought  now  on  a 
3.89%  basis,  the  investment  value  in  1925  will  be  exactly 
1.05  and  there  will  be  neither  profit  nor  loss  in  being  re- 
quired to  surrender  at  1.05;  3.89%  may  be  called  the 
neutral  rate. 

§  288.     Advantageous  Redemption  Ignored 

It  is  necessary  to  bear  in  mind  that  the  higher  the  rate 
of  income  the  lower  is  the  premium ;  if  the  rate  be  more  than 


•  Compare  §  147. 

t  Sprague's  "Extended   Bond  Tables." 

240 


OPTION  OF  REDEMPTION  241 

3.89%,  say  4%,  the  option  may  be  disregarded,  for  we  shall 
surely  have  4%  for  20  years,  and  probably  for  the  full  time. 
In  case  the  rate  of  interest  has  fallen  to  3.89%,  the  issuer 
of  the  bond  may  think  it  advantageous  to  redeem,  so  as  to 
sell  his  new  issue  at  more  than  .05  premium.  Then,  as  our 
bond  stands  at  less  than  1.05,  we  get  a  profit  besides  our  4% 
income.  Thus,  if  the  bond  is  bought  at  a  basis  which  yields 
more  than  3.89%  for  30  years,  we  may  safely  amortize  at 
that  basis  for  20  years,  or  until  the  option  is  exercised. 

§  289.     Disadvantageous  Redemption  Expected 

But  if  the  rate  for  thirty  years,  which  we  may  call  the 
apparent  rate,  or  non-redemption  rate,  is  less  than  3.89%, 
the  bond  will  be  worth  more  than  105  at  the  redemption  date 
and  the  issuer  may  be  expected  to  redeem.  If  he  does  not, 
it  is  because  the  general  rate  of  interest  has  risen  so  that 
he  must  pay  more  than  3.89%,  in  which  case  he  will  allow 
us  to  continue  at  3.89%  till  maturity.  Thus,  if  the  bond  is 
bought  at  a  price  which  would  be  on  an  apparent  basis  of 
less  than  3.89%,  redemption  must  be  expected  as  being 
adverse  to  our  interests.  The  redemption  date  then  becomes 
the  actual  date  of  maturity,  but  the  principal  is  not  1  but 
1.05. 

§  290.    Change  in  Principal 

Let  the  par  be  $100,000  and  the  price  $114,423.38,  which 
is  at  the  apparent  basis  of  3.70%.  To  get  the  actual  basis 
we  must  consider  the  par  as  $105,000  and  the  time  20  years. 
But  if  the  par  is  $105,000,  the  cost  is  not  at  1.1442 1|  but  at 
1.1442||-^1.05  =  1.0897||.  The  cash  rate  is  also  trans- 
formed; the  cash  income  is  still  $4,500,  but  this  is  not  4^2% 
of  $105,000 ;  it  is  only  4  2/7%. 

Therefore,  the  limitation  imposed  by  the  option  of  re- 
demption entirely  changes  the  problem.    Instead  of  a  4%% 


242  PROBLEMS  AND  STUDIES 

bond  for  $100,000,  due  in  30  years,  bought  at  1.1442,  we 
have  a  4  2/7%  bond  for  $105,000,  due  in  20  years,  bought 
at  1.0897. 

No  tables  have  been  published  for  4  2/7%  bonds,  presum- 
ably because  this  exact  case  of  4%%  bonds  redeemable  at 
1.05  is  infrequent.  However,  we  can  easily  construct  them 
by  adding  to  the  value  of  a  4%  bond,  2/7  of  the  difference 
between  a  4%  bond  and  a  5%  bond. 

§  291.     Approximate  Location 

As  a  rough  approximation,  find  1.0897  as  closely  as  pos- 
sible in  the  20-year  tables  for  4%  and  5%  respectively.  The 
nearest  to  1.0897  in  the  4%  table  is  1.08655516,  which  is  a 
3.40%  income;  the  nearest  in  the  6%  table  is  1.08623676,  a 
4.35%  income.  The  required  rate  will  be  about  2/7  of  the 
distance  between  3.40%  and  4.35%. 

4.35  —  3.40=    .95 

2/7  of  .95  =    .27 

3.40+    .27  =  3.67 

Therefore  3.67%  is  the  approximate  rate,  and  we  might 
begin  testing  with  that  rate.  We  notice,  however,  that  the 
approximations  1.08656||  and  1.08624||  are  both  short  of 
1.0897;  hence,  probably  the  rate  will  fall  short  of  3.67%, 
and  it  will  be  easier  to  start  with  the  tabular  rate  3.65%. 
In  fact,  had  we  gone  a  little  further  in  decimals,  using  the 
colored  pages  of  differences  and  sub-differences  in  the  bond 
tables,  we  should  have  obtained  the  following  values  in  the 
4%  table: 

Income  rate,  3.38%,        1.08960122 
Income  rate,  3.37%,        1.09112831 

The  rate  nearest  to  1.0897  in  the  4%  table  is  therefore 
3.38%.     Similarly,   in   the   5%    table   the   nearest   rate   is 


OPTION  OF  REDEMPTION 


243 


4.33% .  Taking  2/7  of  the  difference  between  these  two  rates 
and  adding  this  difference  to  3.38%,  gives  3.65%  as  the 
approximate  income  rate. 

4%  table,  20  years       3.65%     1.0493748 
5%      "        "       "  3.65%     1.1904458 

Difference  .1410710 

1/7  .0201530 

2/7  .0403060 

Add  to  4%  value  1.0493748 

Giving  4  2/7%  value  1.0896808 

This  value  is  very  close  to  1.0897. 

Value  of  $105,000  at  the  same  price.  .$114,416.48 

Actual  price 114,423.38 

Residue 6.90 

This  is  the  nearest  approximation  we  can  obtain  without 
using  more  decimals;  therefore,  3.65%  is  the  actual  rate  of 
income  for  a  4%%  bond  redeemable  at  1.05,  10  years  before 
maturity,  if  purchased  at  114.42,  30  years  before  maturity. 

In  the  diagram  (page  244)  the  dotted  line  marked  3.70  is 
the  apparent  course  of  a  bond  at  114.42,  30  years  to  run ;  but 
the  option  at  105  pulls  it  down  to  a  3.65  basis;  during  the 
last  10  years  it  earns  3.89%,  if  not  redeemed.  The  4%  line, 
as  it  passes  below  the  105  point,  is  unaffected  by  the  option 
of  redemption.  The  issuer  would  not  redeem,  at  105,  a  bond 
whose  value  was  less  than  105. 

To  complete  a  schedule  running  from  the  date  of  issue 
to  that  of  redemption,  we  have  the  following  data : 

Par,  $105,000. 

Cash  interest,  semi-annually,  $2,250,  being  at  the  rate  of 
4  2/7%  per  annum. 

Net  income,  semi-annually,  $1,916.25,  being  3.65%  per 
annum  on  $105,000. 


244 


PROBLEMS  AND  STUDIES 


Difference  of  interest,  $2,250  —  $1,916.25  =  $333.75. 

Present  worth  of  20-year  annuity  of  $333.75  each  half- 
year,  $9,416.48.  Present  value  of  bond  at  3.65%, 
$114,416.48.  Actual  value,  $114,423.38.  Eliminand, 
$6.90. 

We  might  now  proceed  to  amortize  $114,416.48  down  to 
maturity.  Each  term  would  then  have  to  be  corrected  to 
eliminate  the  residue,  $6.90.  The  multiplier  for  this  pur- 
pose would  be : 

9423.38  ^  9416.48  =  1.00073276|  | 

But  we  may  proceed  in  the  other  direction  and  discount 
$333.75  at  various  dates ;  this  has  the  great  advantage  that 


4.1S 

■ 

Graphic  Representation 

OF  THE  Effect  of  aw 

1.14 

N 

V 

Optionai,  Redemption  Date 

%.%» 

• 

N; 

<; 

^ 

'^K 

t.ia 

• 

\ 

*» 

&.X1 

t.xo 

• 

^ 

•^ 

&.O0 

i.oe 

^ 

^ 

\ 

^ 

v 

t*7 

> 

\ 

^^"\^<         ^\\         '  • 

i.e< 

X.05 

1.04 

• 

vet 

\^. 

1*1 

^\ 

VAR  1.©© 

...X. 

-i-l. 

JU 

J. 

Om 

....(..  .  iS«i 

o 


&0 

as 

ao  vsX 

•Z^XUXBMXatK 

SVX  JL»«X> 

Jkvm* 

FAVASCS 

OPTION  OF  REDEMPTION 


245 


$333.75  may  be  first  multiplied  by  1.00073276,  thus  accom- 
plishing the  elimination  process  once  for  all. 

$333.75  X  1.00073276  =  $333.99456 
This  last  is  substituted  as  a  base  in  place  of  $333.75,  and 
we  proceed  to  discount,  using  the  factor  .982077093]  |,  which 
is  the  reciprocal  of  the  semi-annual  ratio  1.01825,  in  the 
tabular  method : 

$333.9946 


294.6231 

29.4623 

2.9462 

8839 

884 


6 


$328.0084  %  year  before  maturity 


294.6231 
19.6415 

7.8566 

79 

4 

$322.1295  1  year  before  maturity 

etc. 

§  292.     Problems  Involving  Optional  Redemption  Dates 

(67)  If  a  4%  bond  is  redeemable  25  years  before 
maturity  at  105,  what  is  the  neutral  rate  of  income? 

(68)  If  a  bond  reads  at  4%,  but  the  amount  which  will 
be  received  is  1.05  of  the  nominal  par,  what  is  the  actual 
percentage  of  cash  income  ? 

(69)  A  50-year  4%  bond  is  redeemable  at  105  after  25 


246  PROBLEMS  AND  STUDIES 

years.    Find  its  actual  income  rate  if  bought  at  (a)  105,  (b) 
106,  (c)  107,  (d)  108,  and  (e)  109. 

(70)  A  30-year  5%  bond  is  redeemable  at  110  after  15 
years.  Find  at  what  price  it  should  be  bought  when  issued 
to  pay  (a)  3.90%,  (b)  4.40%. 

§  293.     Rule  for  Determining  Net  Income 

We  are  now  prepared  to  formulate  a  rule  for  determin- 
ing the  net  income  yielded  at  a  certain  price,  by  a  bond  bear- 
ing a  certain  par  interest  but  subject  to  redemption  at  an- 
other price,  on  the  assumption  that  the  right  will  be 
exercised. 

(1)  Divide  the  nominal  cash  rate  of  interest  by  the  re- 
demption price  per  unit ;  the  quotient  will  be  the  actual  cash 
rate,  consisting  of  a  whole  number  and  a  fraction;  e.g., 
4y2%-^1.05  =  4  2/7%. 

(2)  Divide  the  purchase  price  by  the  same  divisor,  giv- 
ing the  actual  purchase  price  per  unit;  e.g.,  1.1442 -r- 
1.05  ==1.0897. 

(3)  Select  two  different  bond  tables,  one  at  a  lower, 
one  at  a  higher  cash  rate  than  the  actual  rate  obtained  in 
paragraph  1.  These  should  be  even  rates,  not  fractional, 
and  1%  apart.  Find  the  column  for  the  number  of  years 
before  redemption;  e.g.,  4%  and  5%,  20  years. 

(4)  In  each  of  these  columns  find  the  nearest  price  to 
the  actual  purchase  price  in  paragraph  2 ;  e.g.,  in  4%  table, 
1.08656;  in  5%  table,  1.08624. 

(5)  Set  down  the  two  rates  of  net  income  found  op- 
posite these  values,  and  find  their  difference;  e.g.,  inter- 
rates,  4.35%  and  3.40%  ;  difference,  .95%. 

(6)  Take  such  a  fraction  of  the  difference  as  is  shown 
by  the  fractional  part  of  the  mixed  number  which  represents 
the  actual  cash  rate;  add  the  result  to  the  smaller  rate  and 


OPTION  OF  REDEMPTION 


247 


the  sum  is,  approximately,  the  desired  yield;  e.g.,  2/7  X 
.95  =  .27 ;  .27  +  3.40  =  3.67. 

(7)  Try  the  nearest  rates  from  the  table  until  one  is 
found  which  produces  the  desired  price;  e.g.,  3.65  produces 
1.08968. 

§  294.     Answers  to  Problems  Involving  Optional  Redemp- 
tion Dates 

Problem  (67) 

3.69%  + 
Problem   (68) 

3  17/21%,  or  3.80952%  + 

Problem   (69) 

(a)  Between  3.77%  and  3.78%;  (b)  3.73%+; 
(c)  3.69% ;  (d)  3.63% ;  (e)  3.57%  + 
Problem  (70) 

(a)  118.005676;  (b)  109.042757 


CHAPTER  XXVII 

BONDS    AT   ANNUAL   AND   OTHER    RATES 

§  295.     Standard  of  Interest 

In  popular  usage  and,  in  fact,  in  legalized  usage,  though 
not  from  the  mathematical  standpoint,  the  interest  on  a 
given  principal  is  directly  proportional  to  the  time;  that  is, 
iif  the  interest  is  six  dollars  on  a  hundred  for  a  year,  it 
must  for  six  months  be  three  dollars,  and  for  three  months 
one  dollar  and  a  half.  These  three  rates  are  popularly  re- 
garded as  identical,  but  actually  they  are  very  different.  A 
single  standard  should  be  preserved,  and  when  in  any  prob- 
lem "6  per  cent"  is  once  taken  as  meaning  "3  per  cent  per 
half-year,"  it  must  not  be  arbitrarily  shifted  to  mean  "1% 
per  cent  per  quarter,"  which  is  really  "3.0225  per  cent  per 
half-year." 

If  the  ratio  of  increase  or  income  yield  be  kept  at  the 
same  unvarying  standard,  the  frequency  of  collection,  or 
cash  payment,  affects  the  value  of  the  investment.  To  change 
the  coupon  from  half-yearly  to  quarterly,  must  necessarily 
enhance  the  value  of  the  annuity  made  up  of  the  coupons. 
The  nearer  any  one  of  them  approaches  to  the  present,  or 
the  less  time  one  must  wait  for  his  money,  the  more  nearly 
is  it  worth  its  par ;  while  the  present  worth  of  the  principal 
remains  the  same,  unless  we  vary  the  income  yield. 

§  296.     Semi-Annual  and  Quarterly  Coupons 

A  bond  for  $1,000,000,  due  in  one  year,  bearing  semi- 
annual coupons  at  6  per  cent  per  annum,  at  a  price  to  net 

248 


BONDS    AT    ANNUAL    AND    OTHER    RATES      249 

21/2  per  cent  computed  semi-annually  (1%%  per  period), 
is  worth,  according  to  all  tables  and  computations  (except 
the    fictitious    one    of    "reinvestment"    at    an    arbitrary 

rate) $1,034,354.52 

thus, 

Present  worth  of  first  coupon,  $30,000, 

one  period,  1%% ,. $      29,629.63 

Present  worth  of  second  coupon,  $30,- 

000,  two  periods,  114% 29,263.83 

Present  worth  of  principal,  $1,000,- 

000,  two  periods,  1%% 975,461.06 

By  the  method  in  §  111 $1,034,354.52 

Or,  using  the  method  in  §  116,   we  should  take  the 

nominal  interest $30,000 

subtract  from  it  the  effective  interest. .: 12,500 

and  obtain  the  interest-difference $17,500 

An  annuity  of  $17,500,   for  two  terms,  at 

1^A%,  would  be  the  premium. $34,354.52 

The  company  issuing  the  bonds  is  willing  in  return  for 
certain  concessions  to  make  its  interest  payments  quarterly. 
How  much  would  this  add  to  the  value  of  the  bond,  the  in- 
come yield  being  still  2%  per  cent  on  a  semi-annual  basis? 

If  the  bond  be  made  quarterly,  the  same  cash  is  received 
each  half-year,  but  $15,000  of  it  is  received  three  months 
earlier  than  before.  On  this  $15,000  the  bondholder  is  en- 
titled to  only  3  months'  interest,  instead  of  6  months,  at 
1%%  per  half-year;  therefore,  a  quarter's  interest  on  this 
quarterly  coupon  must  be  deducted  each  half-year  from  the 
entire  interest  earned. 

We  must  be  careful,  however,  to  compute  the  interest  cor- 
rectly on  this  advanced  coupon.     It  must  be  at  .00623059, 


250 


PROBLEMS  AND  STUDIES 


not  at  .00625.  Interest  at  a  half-period  is  not  half  of  the 
.0125,  but  the  square  root  of  the  ratio  1.0125,  less  the  1; 
Vi.0125  =  1.0062305911,  interest  =  .00623059.  Otherwise 
we  should  be  using  a  higher  rate  than  1.0125  for  the  half- 
year,  nearly  1.01254.  The  interest  to  be  deducted  each  half- 
year  is  $15,000  X  .00623059  =  $93.46.  The  effective  in- 
terest is  $12,500  — $93.46  =  $12,406.54,  and  the  interest- 
difference  $30,000  —  $12,406.54  =  $17,593.46.  If  we  should 
now  consider  each  instalment  of  the  annuity  to  be  $17,593.46 
instead  of  $17,500,  we  should  have  the  premium  for  quarter- 
ly coupons.  Therefore,  the  two  annuities  (or,  in  other 
words,  the  premiums)  at  any  point  must  be  to  each  other 
as  $17,593.46  :  $17,500;  or  the  ratio  of  the  quarterly 
premium  to  the  semi-annual  is  1.005340507.  Hence  the 
multiplier  .0053405  on  page  VII  of  Sprague's  "Extended 
Bond  Tables.'' 

In  symbols,  the  income  rate  becomes  (instead  of  i), 
i —  f  (Vl  +  ^*  —  1)  and  the  interest-difference  becomes  (in- 
stead of  c  —  i)y  c—  [i  — |(VlTT— 1)]  =c  —  i+^ 
(VTTl  — 1),  which  divided  by  (c  —  i)  gives  the  propor- 
tion 1-f  ^  (Vl  +  ^  — 1)  ^  ^.^  ^^^  ^^^^  ^^^^^  1.0053405. 
c  —  i 

The  process  of  finding  .0053405  may  be  briefly  expressed 
thus: 

Rule :  Divide  a  quarter's  interest  on  a  quarterly  coupon 
by  the  interest-difference. 

The  value  of  the  bond  when  trimestralized  (reduced  to  a 
quarterly  basis)  is,  therefore : 

At  semi-annual  payments .$1,034,354.52 

Added    for    quarterly    coupon,    34,354.52  X 

.0053405 183.47 

Value  trimestralized $1,034,537.99 


BONDS  AT  ANNUAL  AND  OTHER  RATES 


251 


This  may  be  tested  by  multiplying  down  to 
maturity,  3  months  at  a  time,  viz. : 

$1,034,537.99  X  .00623059  +  6,M5.Y9 


$1,040,983.78 
— 15,000.00 


$1,025,983.78 
$1,025,983.78  X  .00623059  +  6,392.48 


$1,032,376.26 
— 15,000.00 


$1,017,376.26 
$1,017,376.26  X  .00623059  +  6,338.86 


$1,023,715.12 
— 15,000.00 


$1,008,715.12 
$1,008,715.12  X  .00623059  +  6,284.88 


$1,015,000.00 
Final  payment,         1,015,000.00 


We  will  now  take  an  example  where  the  effective  rate  is 
greater  than  the  cash  rate.  A  bond  of  $1000  at  4%  (sem.), 
due  in  ten  years,  is  bought  so  as  to  give  a  net  income  of  5% 
(sem.)  ;  what  will  be  its  value  if  trimestralized ? 

The  normal  or  semi-annual  value  is  by  all  tables.. $922,054 

The  discount,  -^-^  X  ( '^ ,,   }  .,^^ ,  is 77.946 

I  V  (1  +  0  / 

The  multiplier*  is 0248457 

77.946  X  .0248457,  amount  of  added  value,  =  .  .  .$1.93661 

922.054  +  1.937  = 923.991 


Spragrue's  "Extended  Bond  Tables,"  page  VII. 


252 


PROBLEMS  AND  STUDIES 


§  297.     Shifting  of  Income  Basis 

This  is  the  correct  vakie,  the  income  basis  being  un- 
changed. But  in  some  recent  books  we  find  the  quarterly- 
value  io  be  stated  as  $921,683,  which  is  a  surprising  result, 
for  we  should  not  expect  the  value  of  the  security  to  be 
diminished  by  a  more  frequent  interest-payment.  The 
trouble  is,  that  the  income  basis  has  been  suddenly  shifted 
from  4%  semi-annual  to  4%  quarterly,  and  we  are  given 
comparisons  between  the  following  values : 

(a)  At  4%  semi-annual  basis,  coupon  5%  semi-annual. 

(b)  At  4%  quarterly  basis,  coupon  5%  quarterly. 
Whereas  the  value  really  desired  is : 

(c)  At  4%  semi-annual  basis,  coupon  5%  quarterly. 

In  all  the  tables  using  the  basis  (b),  the  values  below  par 
are  all  apparently  diminished  by  frequency  of  payment.  The 
author's  tables  are  computed  on  the  semi-annual  income 
basis,  though  the  coupons  may  be  quarterly  or  annual. 

§  298.    Problems — Bonds  at  Varying  Rates 

(71)  A  5%  quarterly  bond  for  $100,000  has  5  years 
to  run  on  a  4%  semi-annual  basis;  what  is  its  value? 

(72)  Ascertain  the  value  of  the  same  bond  at  4I/2  years. 

(73)  Derive  the  4%  years'  value  from  the  5  years,  and 
obtain  the  same  value  as  in  (72). 

(74)  Find  the  value  of  a  2%  quarterly  bond,  5  years  to 
run,  which  nets  1.80%  semi-annually. 

(75)  Two  issues  of  20  year,  31/^  %  bonds,  each 
$100,000,  are  offered;  one  with  interest  semi-annually  at 
95.29,  the  other  quarterly  at  95.38 ;  find  the  better  purchase. 

(76)  Which  is  the  better  purchase : 

$1,000,000    4%'   quarterly   bonds,    10   years,    at 

104.33,  or 
$1,000,000  3%  semi-annual  bonds,  10  years,  at 

95.50? 


BONDS    AT    ANNUAL    AND    OTHER    RATES      253 

§  299.    Answers  to  Problems — Bonds  at  Varying  Rates 

Problem   (71) 

$104,608.02 
Problem  (Y2) 

$104,182.64 
Problem  (73) 

Value  at  6  years , $104,603.02 

Of  this   $1,250   is   payable   in   three 
months. 

Present  worth  at  4%  semi-annually. .       1,237.69 

The  remainder $103,365.33 

Produces  income  at  .02 2,067.31 

$105,432.64 
Cash  interest  received 1,250.00 

Value  at  4%  years > $104,182.64 

An  alternative  solution  for  this  problem,  and  the  one 
usually  employed,  is  as  follows : 

Value  at  5  years. $104,603.02 

This  multiplied  by  the  quarterly  effective 
rate,  .00995049  (which  is  the  square 
root  of  1.02,  less  1)  gives 1,040.85 

$105,643.87 
Less  quarterly  coupon 1,250.00 

Giving  value  at  ^%  years $104,393.87 

This  multiplied  by  .00995049  gives  the  next 

quarterly  income 1,038.77 

$105,432.64 
Less  quarterly  coupon. 1,250.00 

Giving  value  at  4%  years. ,. . ... . ..  .$104,182.64 


254 


PROBLEMS  AND  STUDIES 


Problem   (74) 

$100,973.61 
Problem  (75) 

The  quarterly  bonds. 
Problem  (76) 

The  semi-annual  bonds. 

§  300.     Bonds  with  Annual  Interest — Semi-Annual  Basis 

Bonds  on  which  the  interest  is  paid  only  once  a  year  are 
somewhat  rarer  than  those  where  it  is  paid  four  times  a 
year;  but,  when  they  do  occur,  means  should  be  provided 
for  ascertaining  their  value  at  any  given  rate  reduced  to  the 
standard  of  semi-annual  income.  This  is  somewhat  easier 
than  finding  the  value  of  a  quarterly  bond  on  a  semi-annual 
income  basis. 

We  may  begin  by  a  simple  example  using  the  discount 
method,  either  by  division  or  by  multiplication,  taking  a  4% 
annual  bond  yielding  3%  semi-annually,  2  years  to  run,  for 
$100,000. 

Beginning  at  maturity  at  par. .....' $100,000.00 

and  adding  to  it  the  annual  coupon  then  due. .       4,000.00 


$104,000.00 


We  discount  this  by  dividing  by  the  ratio,  1.015, 
or,  what  is  the  same  thing,  multiplying 
by  its  reciprocal,  .98522167;  $104,000 -f- 
1.015  or  X  .98522167  = $102,463.05 


This  is  the  value,  flat,  6  months  before  maturity. 
If  there  were  a -payment  of  interest  at  this 
date  we  should  add  its  value.  But  there  is 
none;  hence  we  continue  the  process,  $102,- 
463.05  -^  1.015  or  X  .98522167  = $100,948.82 


BONDS  AT  ANNUAL  AND  OTHER  RATES 


255 


Here  we  add  the  coupon  payable  one  year  before 

maturity 4,000.00 

$104,948.82 

We  discount  this  for  another  half-year,  $104,- 

948.82  -^  1.015. $103,397.85 

and  again,  $103,397.85  ^  1.015 $101,869.81 

which  is  the  value  required.  '  = 

To  test  this,  let  us  multiply  down  to  maturity : 

Value  at  2  years $101,869.81 

Income  at  1%%,  %  year 1,018.70 

509.35 

Value  at  1%  years,  flat , $103,397.86 

No  coupon. 

Income  at  1%%,  %  year 1,033.98 

516.99 

$104,948.83 
Annual  coupon  paid 4,000.00 

Value  at  1  year .$100,948.83 

Add  1/2  year's  income,  at  11/2% 1,009.48 

504.74 

$102,463.05 

Add  last  %  year's  income,  at  1%%  •  • 1,024.63 

512.32 

Total  principal  and  interest $104,000.00 


§  301.    Annualization 

We  will  now  annualize  the  above  process;  that  is,  in- 
stead of  multiplying  twice  by  1.015,  we  will  multiply  once 
by  1.030225,  which  is  1.015  X  1.015,  or  (1.015)^ 


256  PROBLEMS  AND  STUDIES 

As  before,  beginning  with $101,869.81 

we  multiply  by  1.030225. 3,056.09 

20.37 

2.04 

.61 

$104,948.82 
and  subtract  the  coupon 4,000.00 

$100,948.82 

again  multiply  by  1.030225 3,028.47 

20.19 

2.02 

.50 


giving  the  same  result $104,000.00 

Thus,  income  has  been  received  on  all  of  the  investment 
outstanding  at  1.5%  per  half-year,  or  at  3.0225%  per  year. 

§  302.     Semi-Annual  Income  Annualized 

Suppose  now  that  we  take  the  case  of  an  ordinary  half- 
yearly  bond  paying  a  cash  interest  of  2%  twice  a  year,  and 
yielding  1.5%  half-yearly,  with  the  purpose  of  annualizing 
in  this  case  also.  The  ratio,  when  annualized,  is  the  same 
as  before,  1.030225,  but  there  are  two  semi-annual  coupons 
of  $2,000.00  each,  instead  of  the  single  annual  coupon  of 
$4,000.00  as  in  the  previous  case.  The  first  of  these  cou- 
pons, if  deferred  to  the  end  of  the  year,  will  increase  at 

the  semi-annual  ratio  of  1.015  to $2,030.00 

The  second  coupon  remains 2,000.00 

The  entire  cash  interest,  when  concentrated  at  the 

end  of  the  year,  is  therefore  equivalent  to $4,030.00 

The  processes  of  multiplying  down  to  maturity,  using 
both  semi-annual  and  annual  periods,  are  shown  below  side 
by  side,  beginning  with  the  value  $101,927.19  found  from 


BONDS  AT  ANNUAL  AND  OTHER  RATES 


257 


tables  or  by  computation.  In  a  third  column  appears  the  an- 
nuaHzed  process  in  the  case  of  a  4%  annual  coupon.  In  all 
three  cases,  the  net  income  is  1.5%  per  half-year,  or  its 
equivalent,  3.0225%  annually. 


Cash  Interest 
2%  PER  Half- Year 

Cash  Interest 
4%  per  Year 

Ordinary   Process 

Annualized  Process 

Annualized  Process 

$101,927.19 

1,019.27 

509.64 

$101,927.19 

3,057.82 

20.38 

2.04 

.51 

$101,869.81 

3,056.09 

20.37 

2.04 

.51 

$103,456.10 
2,000.00 

$101,456.10 

1,014.56 

507.28 

' 

$102,977.94 
2,000.00 

$105,007.94 
4,030.00 

$104,948.82 
4,000.00 

$100,977.94 

1,009.78 

504.89 

$100,977.94 

3,029.34 

20.20 

2.02 

.50 

$100,948.82 

3,028.47 

20.19 

2.02 

.60 

$102,492.61 
2,000.00 

$100,492.61 

1,004.93 

502.46 

$102,000.00 

$104,030.00 

$104,000.00 

258 


PROBLEMS  AND  STUDIES 


§  303'     Comparison  of  Annual  and  Semi-Annual  Bonds 

In  each  of  these  columns  the  proper  principal  is  attained 
at  maturity,  together  with  its  accompanying  interest,  either 
actual  or  annualized.  Observing  the  first  and  second 
columns,  we  see  that  a  semi-annual  4%  bond  is  effectively 
a  4.03%  annual  bond,  the  net  income  in  both  cases  being 
3.0225%  per  annum.  Comparing  the  second  and  third 
columns,  the  point  to  be  noted  is  that  their  chief  difference 
lies  in  the  effective  cash  rates,  one  being  4.03%  and  the 
other  4%  ;  in  the  semi-annual  bond,  annualized,  the  interest- 
difference  between  the  cash  and  income  rates  is 

$4,030.00  —  $3,022.50  =  $1,007.50 
In  the  annual  bond,  it  is..  4,000.00  —    3,022.50=       977.50 

§  304.     Finding  Present  Worth  of  an  Annuity 

These  interest-differences,  $1,007.50  and  $977.50,  are 
important  because  (according  to  the  second  rule  in  Chapter 
X)  we  have  only  to  multiply  these  two  interest-differences 
by  the  present  worth  of  an  annuity  of  $1  for  2  periods  at 
3.0225%,  in  order  to  obtain  the  respective  bond  premiums. 
We  might  find  this  present  worth  approximately  from  Table 
IV*  by  interpolation  between  the  3%  and  31/2%  columns, 
but  a  much  more  accurate  result  may  be  obtained  by  the 
use  of  Table  II*,  where  we  can  find  the  present  worth  of  $1 
for  4  periods  at  1.015,  which  is  exactly  equivalent  to  the 
present  worth  of  $1  for  2  periods  at  1.030225. 

This  value $  .94218423 

must,  according  to  Chapter  V,  be  subtracted 

from 1.00000000 

and  the  remainder $  .05781577 

must  be  divided  by  the  income  rate 030225 

The  quotient  is $1.9128460 

*  In  Chapter  XXXII, 


BONDS    AT    ANNUAL    AND    OTHER    RATES 


259 


which  is  the  present  value  of  an  annuity  of  $1  for  2  periods 
at  3.0225%  per  period.  The  foregoing  is  an  application  of 
the  two  symbolic  rules,  D  =  1  —  p  and  P  =  D  -v- 1. 

In  order  to  obtain  the  bond  premiums,  we  must  multiply 
the  above  present  worth  by  1,007.50  in  the  case  of  the  semi- 
annual bond,  and  for  the  annual  bond  by  977.50. 
Premium  on  semi-annual 

bond $1.912846  X  1,007.50  =  $1,927,192 

Premium  on  annual  bond  1.912846  X     977.50  =    1,869.807 

These  premiums  agree  perfectly  with  the  values  pre- 
viously obtained  otherwise,*  viz. :  $101,927.19  and  $101,- 
869.81. 

As  another  example,  take  that  of  a  4%  annual  bond 
yielding  5%,  for  two  years.  Evidently  this  will  be  at  a 
discount  instead  of  at  a  premium.  To  annualize  the  ratio 
1.025,  multiply  it  by  itself,  giving  1.050625 ;  the  annualized 

interest  rate  is  therefore : .  .     .050625 

from  this  subtract 04 


giving  as  the  interest-difference 010625 

To  find  the  present  worth  of  an  annuity  of  $1  for  2  (an- 
nual) periods  at  5.0625%,  take  from  Table  II,*  column 
2%%,  the  value  for  4  (semi-annual)  periods.  .$  .90595064 
subtract  from 1.00000000 

The  compound  discount  is  therefore $  .09404936 

Divide  by  .050625 ;  the  quotient  is $1.8577652 

which  is  the  required  present  worth  of  an  an- 
nuity of  $1  for  2  periods  at  5.0625%. 
Multiply    this    by    .010625;    $1.8577652||  X 

.010625  = $  .01973876 

This  is  the  discount,  which,  subtracted  from  par  1.00000000 

gives  the  value  of  a  $1  bond, $  .98026124 

*  In  Chapter  XXXII, 


26o  PROBLEMS  AND  STUDIES 

This  may  be  tested  by  multiplying  down  to  maturity : 

X  1.025  .02450653 


X  1.025 

$1.00476777 
.02511919 

$1.02988696 
.04 

X  1.025 

$  .98988696 
.02474718 

X  1.025 

$1.01463414 
.02536586 

$1.04000000 
.04 

$1.00 

§  305.     Rule  for  Bond  Valuation 

We  are  now  prepared  to  formulate  a  rule  for  valuing  an 
annual  bond  on  a  semi-annual  basis  without  reference  to 
the  values  of  a  corresponding  ordinary  (or  semi-annual) 
bond. 

Rule  1: 

(a)  Annualize  the  rate  of  interest  (find  the  equivalent 
annual  income  rate) ;  e.g.,  1.015^  =  1.030225. 

(b)  Subtract  this  rate  from  the  annual  coupon,  or  vice 
versa,  to  give  the  interest-difference;  e.g.,  .04  —  .030225  = 
.009775. 

(c)  Multiply  the  latter  by  the  present  worth  of  an  an- 
nuity of  $1  for  the  number  of  annual  periods  at  the  an- 
nualized rate,  giving  the  premium  or  the  discount;  e.g., 
.009775  X  1.9128453  =  .0186981. 

Where  the  values  of  the  ordinary  semi-annual  bond 


BONDS    AT    ANNUAL    AND    OTHER    RATES      261 

have  already  been  calculated,  as  in  the  bond  tables,  it  will 
be  possible  to  obtain  therefrom  the  values  of  the  annual 
bond,  with  a  saving  of  time. 

§  306.     Multipliers  for  Annualizing 

For  each  combination  of  a  cash  rate  with  an  income  rate, 
a  multiplier  may  be  found  which,  applied  to  the  premium  or 
the  discount  for  any  number  of  years  on  a  semi-annual  bond, 
will  give  the  depreciation  caused  by  the  collection  of  the 
interest  once  a  year  only;  and  this  multiplier  will  be  con- 
stant, whatever  the  time.  A  table  of  these  multipliers  will 
be  found  in  Spragne's  "Extended  Bond  Tables,"  page  VIII. 

In  the  example  given  in  §  300  we  have  a  4%  annual 
bond  yielding  3%  semi-annually.  On  page  VIII*  in  the 
column  headed  "4%  Bond"  on  the  line  opposite  "3%"  is 
the  multiplier  .0297767.  The  premium  on  the  ordinary 
semi-annual  bond  for  $100,000  at  2  years,  we  have  seen,  is 
$1,927.19. 

$1,927.19  X  .0297767  = $  57.385 

As  the  value,  if  semi-annual,  would  be.  .  .    101,927.192 
the  value  of  the  annual  bond  is  reduced  to  101,869.807 

In  the  example  in  §  304,  the  annualizer,  or  multiplier,  for 
a  4%  bond  to  yield  5%  is  found  from  the  table 

to  be ,. .      .0493827 

The  value  of  a  semi-annual  bond  of  $1  at  2 

years   is $.98119013 

or  its  discount  is ,. 01880987 

$.01880987  X  .0493827  = 00092888 

which  subtracted  from 98119013 

gives  the  annualized  value 98026125 

This  differs  from  the  one  already  given 98026124 

by  1  cent  on  a  million  dollars,  owing  to  decimals  having 
been  rounded  off. 

*  Sprague's  "Extended  Bond  Tables." 


262  PROBLEMS  AND  STUDIES 

These  multipliers  are  obtained  by  the  following  formula, 
in  which  c  and  i  represent  the  nominal  rates  per  annum. 

ci 

(4  +  i)  (c-i) 

§  307.     Formula  for  Annualizer 

The  formula  may  be  thus  expressed  as  a  rule. 
Rule  2 :  To  find  the  annualizer  for  any  two  rates : 

(a)  Multiply  the  rates  together  for  a  dividend;  e.g., 
.04  X  .03  =  .0012. 

(b)  Multiply  4  +  the  income  rate,  by  the  difference  of 
rates  for  a  divisor;  e.g.,  4.03  X  .01  =  .0403. 

(c)  Their  quotient  will  be  the  required  multiplier,  or  an- 
nualizer; e.g.,  .0012  ^  .0403  ==  .029776675. 

The  product  of  the  premium  by  the  annualizer  is  always 
subtracted  from  the  semi-annual  value;  and  sometimes  the 
resulting  value  may  be  shifted  to  a  discount  from  a  premium, 
even  if  it  was  a  premium  which  was  extracted  from  the 
table.  Thus,  in  the  case  of  a  $1,000,000  5%  annual  bond, 
payable  in  one  year,  netting  4.95%,  the  premium  $482.03  X 
the  annualizer  1.22237313  =  $589.22,  and  the  value  of  the 
annual  bond  becomes  $999,892.81. 

It  must  be  observed  that  only  values  for  full  years  can 
be  obtained  in  either  of  these  ways.  An  odd  half-year  is  a 
"broken"  period,  and  must  be  treated  as  in  Chapter  XL 

§  308.     Conventional  Process 

While  the  foregoing  is  the  method  which  would  doubt- 
less be  followed  in  buying  and  selling,  a  more  accurate  re- 
sult, from  a  mathematical  standpoint,  would  be  obtained  by 
using  as  the  half-year  value  the  one  found  by  multiplying 
down  at  the  effective  rate. 

Thus,  in  a  bond  at  4%,  payable  annually,  on  a  3%  semi- 
annual basis,  the  values  are : 


BONDS    AT    ANNUAL    AND    OTHER    RATES      263 

2  years  before  maturity. .  .$1,018,698.07 
lyear  "  "  ...  1,009,488.22 
Maturity 1,000,000.00 

The  amortization  for  the  first  year  is  $9,209.85,  and  for 
the  second  $9,488.22.  Halving  these  severally,  the  values 
by  half-years  appear  as  follows : 

Values  Di 

2       years   $1,018,698.07     $4,604.92 

11/2  years   1,014,093.15       4,604.93 

1       year     1,009,488.22       4,744.11 

1/2  year     1,004,744.11       4,744.11 

Maturity 1,000,000.00 

§  309.     Scientific  Process 

The  foregoing  result  would  be  in  accordance  with  the 
conventionally  established  rule  that  during  any  period 
(which  is  here  a  year)  simple  interest  must  prevail  and  the 
amortization  accrue  proportionately  to  the  time  elapsed  from 
the  beginning  of  the  period. 

But  the  half-year  may,  with  equal  propriety,  be  con- 
sidered the  period,  since  the  income  is  on  a  semi-annual 
basis.    Under  this  assumption  we  must  multiply  down : 

Value,  2  years $1,018,698.07 

X  1.015 10,186.98 

5,093.49 

Value,  11/2  years $1,033,978.54  flat 

Less  accrued  interest 20,000.00 

Value,  11/2  years $1,013,978.54  and  interest 

Similarly,  the  value  at  one-half  year  is  fixed  at  $1,004,- 
630.54,  and  the  series  with  differences  will  appear  as 
follows : 


264  PROBLEMS  AND  STUDIES 

Values  Di 

2       years  $1,018,698.07  $4,719.53 

11/2  years    1,013,978.54  4,490.32 

1       year     1,009,488.22  4,857.68 

1/2  year     1,004,630.54  4,630.54 

Maturity 1,000,000.00 

In  the  second  half  of  each  year  there  is  less  amortization, 
and  consequently  more  earning  than  in  the  first  half;  but 
this  may  be  defended  on  the  ground  that  by  the  conditions 
prescribed,  interest  is  compounded  semi-annually.  The 
earning  power  at  compound  interest  must  continue  to  in- 
crease until  a  cash  payment;  and  there  is  no  cash  payment 
at  the  mid-year. 

§  310.     Values  Derived  from  Tables 

This  latter  form  of  valuation  at  mid-years  is  recom- 
mended for  comparative  (non-commercial)  purposes. 

The  values  at  ^1/2  years,  $1,004,630.54,  $1,013,978.54, 
etc.,  may  be  deduced  from  the  ordinary  extended  tables  by 
multiplying  by  the  annualizer,  with  this  proviso :  that  the 
interest-difference  must  first  be  temporarily  added  to  the 
tabular  premium  or  discount  before  multiplying.  Thus,  in 
the  case  just  considered,  the  excess  of  .02  over  .015  is  .005 
each  half-year ;  or,  on  $1,000,000,  $5,000. 

To  find  the  value  for  1%  years,  take  from  the 

table  the  premium.  . $14,561.00 

add  the  interest-difference 5,000.00 

giving  the  multiplicand $19,561.00 

which,   multiplied   by  the   annualizer   .9702233, 

equals $18,978.54 

from  which  again  subtract. 5,000.00 

giving  the  premium  as  above $13,978.54 


BONDS    AT    ANNUAL    AND    OTHER    RATES      265 

§  311.     Successive  Process 

In  general,  when  a  schedule  is  to  be  formed  for  an  an- 
nual or  a  quarterly  bond,  on  a  semi-annual  basis,  it  will  be 
found  easier  after  ascertaining  the  initial  value  to  multiply 
down  to  maturity,  as  that  will  usually  require  fewer  figures. 

§312.     Problems  and  Answers — Varying  Time  Basis 

(Y7)  $25,000  4%  bonds,  interest  payable  annually,  8 
years  to  run ;  what  is  the  price  at  a  3.70%  semi-annual  basis? 

(78)  What  multiplier  will  annualize  the  premium  on 
the  above  bonds  as  given  in  the  regular  bond  table  ? 

(79)  An  offering  is  made  of  $30,000  31/2%  bonds,  in- 
terest payable  annually,  of  which  $10,000  mature  in  one 
year  and  $10,000  each  year  thereafter.  What  should  be 
paid  for  them  to  produce  3.40%  semi-annually? 


Answers : 
Problem  (77) 

$25,452.30 
Problem   (78) 

.8777970411 
Problem   (79) 

$30,040.34 

§  313.     Bonds  at  Two  Successive  Rates 

Occasionally  bonds  are  issued  with  the  agreement  that 
the  interest  paid  shall  be  at  a  certain  rate  for  some  years, 
and  at  another  rate  for  the  remainder  of  the  time  to  maturity. 
An  example  is  a  fifty-year  bond  bearing  4%  for  20  years 
and  5%  for  the  following  30  years.  The  problem  is  then  to 
find  the  price  at  which  they  will  pay  a  certain  income,  say 
3.60%. 

Each  of  the  two  successive  cash  rates  will  cause  a 
premium,  and  we  may  calculate  these  premiums  separately. 


266  PROBLEMS  AND  STUDIES 

§  314.     Calculation  of  Immediate  Premium 

The  premium  caused  by  the  4%  rate  will  last  only  20 
years  and  will  then  vanish ;  hence,  this  premium  is  just  the 
same  as  that  on  a  plain  4%  bond  for  20  years,  netting 
3.60%,  which  we  find  by  calculation  or  from  tables  to  be 
$56,680.10  on  $1,000,000. 

§315.     Calculation  of  Deferred  Premium 

The  premium  produced  by  the  5%  rate  does  not  take 
effect  immediately,  but  after  20  years.  It  is  a  deferred  an- 
nuity. An  annuity  for  the  entire  50  years  of  the  excess  in- 
terest, 1.40%,  or  in  other  words  the  premium  on  a  fifty 

year  5%  bond  to  net  3.60%,  is $323,568.65 

But  during  the  first  20  years  there  will  be  no 
such  premium ;  we  have  already  charged  that  at 
4%.  Hence  we  must  by  subtraction  eliminate  the 
analogous  5%  premium  for  20  years,  which  is  198,380.36 

leaving  a  remainder $125,188.29 

which  is  the  premium,  or  present  worth,  of  the  enjoyment  of 
a  5%  cash  rate  (as  against  a  3.60%  income  rate)  commenc- 
ing 20  years  from  date  and  continuing  till  50  years  from 
date. 

Adding  together  the  two  premiums,  $56,680.10  and 
$125,188.29,  we  have  $181,868.39  as  the  premium  which 
should  be  paid  for  the  bond. 

A  simpler  way  to  apply  the  principle  is  to  add  together 

the  4%  value  for  20  years $1,056,680.10 

and  the  5%  value  for  50  years 1,323,568.65 

$2,380,248.75 
and  subtract  the  5%  value  for  20  years 1,198,380.36 

giving  the  value  of  the  composite  bond $1,181,868.39 


BONDS    AT    ANNUAL    AND    OTHER    RATES     267 

This  procedure  has  the  advantage  that  it  applies  alike  to 
bonds  which  are  selling  at  a  premium  and  to  those  which  are 
selling  at  a  discount  and  automatically  allows  for  that  dis- 
tinction. 

§316.     Symbols  and  Rule 

We  may  for  convenience  represent  the  earlier  rate  by  Ci 
and  the  latter  rate  by  C2,  i  being  the  net  income.  We  may  put 
m  for  the  number  of  years  at  which  the  rate  Ci  prevails,  and 
n  for  the  number  of  years  at  C2;  m  +  n  is  the  entire  time. 
The  rule  will  then  be  as  follows : 

Rule :  To  find  the  value  of  a  bond  to  yield  i  per  cent, 
when  by  its  terms  it  pays  cash  interest  at  the  rate  Ci  for  m 
years  and  thereafter  at  Cz  for  n  years,  maturing  in  w  +  w 
years.  Add  together  the  value  of  a  Ci  bond  for  m  years 
and  that  of  a  Cz  bond  for  m-\-  n  years,  and  from  the  sum 
subtract  the  value  of  a  Cz  bond  for  m  years. 

An  example  of  a  bond  of  very  early  maturity  will  il- 
lustrate the  principle  of  the  rule  and  will  admit  of  demonstra- 
tion by  multiplying  down.  A  bond  for  $100,000  paying 
5%  for  1  year  (2  periods)  and  6%  thereafter  for  1%  years 
(3  periods)  is  to  be  valued  so  as  to  yield  the  annual  return 
of  4%. 

§  317.     Analysis  of  Premiums 

If  the  rate  on  the  bond  were  5%  for  the  entire  2%  years, 
its  value,  according  to  the  bond  tables,*  would  be 
$102,356.73.  On  the  other  hand,  if  the  rate  were  6%,  its 
value  would  be  $104,713.46.  Let  us  analyze  these  premiums 
into  their  component  parts,  which  are  the  present  worths  of 
excess  interest  for  five  periods,  $500  per  period  in  the  case 
of  the  5%  bond,  and  $1,000  per  period  in  the  case  of  the 
6%  bond. 


Sprague's   "Extended   Bond  Tables. 


268 


PROBLEMS  AND  STUDIES 


%  year 

1  year 

Premium  one  year  before  maturity 
1%  years 

2  years 
2%  years 


5% 

$490,196 

480.584 


6% 
$980,392 
961.168 


$970.T80  J  $1,941,560 


471.161 
461.923 
452.866 


942.322 
923.846 
905.732 


Premium  2%  years  before  maturity  $2,356,730     $4,713,460 

Any  premium  is  the  sum  of  a  certain  number  of  present 
worths  of  $500  or  of  $1,000.  But  in  the  double-rate  bond, 
the  only  present  worths  that  have  an  influence  on  the  in- 
augural value  are  the  first  two  in  the  5%  column  and  the 
last  three  in  the  6%  column,  as  indicated  by  the  braces  placed 
opposite  them. 

It  is  evident  that  the  values  producing  premiums  at  the 
5%  rate  amount  to  $970.78,  and  that  those  in  the  6% 
column  amount  to  $2,771.90  (the  easiest  way  to  obtain  this 
latter  amount  being  to  subtract  $1,941.56  from  $4,713.46). 
Hence  the  premium  is : 

$970.78  +  $2,771.90  =  $3,742.68 

The  equivalent  process  by  the  rule  would  be : 

Value  of  Cr  bond,  m  years.  .  .$100,970.78 
plus  Value  of  d  bond,  m-\- n  years  104,713.46 


$205,684.24 
less  Value  of  Ca  bond,  m  years. .  .   101,941.56 


$103,742.68 


It  will  be  interesting  to  multiply  down  to  maturity  and 
thus  test  this  result : 


BONDS    AT    ANNUAL    AND    OTHER    RATES      269 

$103,742.68 
+         2,074.85 


$105,817.53 

—  2,500.00 

$103,317.53 
+         2,066.35 

$105,383.88 

—  2,500.00 

$102,883.88 
+         2,057.68 

$104,941.56 

—  3,000.00 

$101,941.56 
+         2,038.83 

$103,980.39 

—  3,000.00 

$100,980.39 
+         2,019.61 

$103,000.00 

—  3,000.00 

'Par        $100,000.00 


It  will  sometimes  be  the  case  that  in  multiplying  down 
the  values  will  increase  for  a  time  and  then  begin  to  de- 
crease at  the  change  of  rate ;  or  vice  versa,  the  values  will  at 
first  decrease  and  then  later  increase. 


270  PROBLEMS  AND  STUDIES 

§  318.     Problems  and  Answers — Successive  Rates 

(80)  An  issue  of  bonds  matures  on  Jan.  1,  1966.  In- 
terest is  to  be  at  6%  till  Jan.  1,  1936,  and  thereafter  at  Q'/o, 
What  is  the  price  at  a  3.60%  basis  on  July  1,  1916? 

(81)  $10,000  of  Waterworks  Bonds,  5  years  to  run, 
first  3  years  at  4%,  thereafter  at  5%;  find  the  value  to 
yield  4.40%. 

(82)  Find  the  value  of  the  same  bonds  to  net  4%%; 
51/4%. 


Answers : 
Problem  (80) 

$1,413,422.66 
Problem  (81) 

$9,988.49 
Problem  (82) 

$9,824.98 ;  $9,617.04 


CHAPTER  XXVIII 

REPAYMENT  AND   REINVESTMENT 

§319.    Aspects  of  Periodic  Payment 

When  a  loan  is  payable  in  equal  periodic  instalments, 
each  covering  the  interest  and  part  of  the  principal,  the 
most  obvious  way  of  looking  at  it  is  that  the  principal  is 
gradually  paid  off;  and  then  we  have  this  aspect: 

(1)  A  diminishing  principal; 

A  diminishing  interest  charge,  and  therefore 

An  increasing  repayment. 
But  precisely  the  same  result  may  be  obtained  from  a 
different  point  of  view  by  assuming  that  no  payment  is 
made  at  all  until  the  final  date  of  maturity,  at  which  time  the 
sinking  fund,  or  sum  of  instalments  plus  interest,  is  just 
sufficient  to  pay  off  the  whole  debt.  In  this  case,  we  will 
have  the  following  aspect : 

(2)  An  unchanged  principal; 
A  uniform  interest  charge ; 

A  uniform  instalment  devoted  to  reinvestment  and 
allowed  to  accumulate. 
As  an  illustration  of  the  first  aspect,  suppose  we  consider 
a  debt  of  $1,000,  bearing  interest  at  3%  per  period.  This 
debt  may  be  extinguished  in  four  periods  by  uniform  instal- 
ments of  $269.03  at  the  end  of  each  period,  as  we  have 
already  pointed  out  in  Chapter  VII.  For  convenience,  how- 
ever, we  again  set  forth  the  details  on  th^  following  page : 

271 


2^2 


PROBLEMS  AND  STUDIES 


Instalment 

Interest 
on  Balance 

Payment  on 
Principal 

Principal 
Outstanding 

$1,000.00 

(1)          $269.03 

$30.00 

$239.03 

760.97 

(2)            269.03 

22.83 

246.20 

514.77 

(3)            269.03 

15.44 

253.59 

261.18 

(4)            269.03 

7.85 

261.18 

0. 

Total,  $1,076.12 

$76.12 

$1,000.00 

Here  we  see  the  diminishing  principal,  the  diminishing 
interest  charge  and  the  increasing  repayment  or  amortiza- 
tion. 

From  the  reinvestment  point  of  view,  we  have : 


Instalment 

Interest  on 
Entire  Principal 

Carried  to 
Sinking  Fund 

Principal 

$1,000.00 

(1) 

$269.03 

$30.00 

$239.03 

1,000.00 

(2) 

269.03 

30.00 

239.03 

1,000.00 

(3) 

269.03 

30.00 

239.03 

1,000.00 

(4) 

269.03 

30.00 

239.03 

1,000.00 

For 
Reinvestment 

Interest  on 
Previous  Total 

Total 
Accumulated 

(1) 

(2) 
(3) 
(4) 

$239.03 
239.03 
239.03 
239.03 

$  7.17 
14.56 
22.15 

$    239.03 

485.23 

738.82 

1,000.00 

The  amortization  of  principal  in  its  two  aspects  as  re- 
payment and  reinvestment  should  be  carefully  studied  and 


REPAYMENT  AND  REINVESTMENT  273 

the  problems  in  connection  with  Chapter  VII  should  be 
worked  over  into  schedule  form  in  each  aspect. 

§  320.     Integration  of  Original  Debt 

This  principle  will  be  found  to  hold :  The  "principal  out- 
standing" by  the  first  method  +  the  "total  accumulated"  by 
the  second  method  =  the  original  debt. 

The  first  point  of  view  is  based  entirely  on  facts.  With- 
out regard  to  reinvestment,  it  is  certain  that  the  borrower 
pays  and  the  lender  receives  the  exact  rate  of  interest  stipu- 
lated for  each  period  on  the  actual  balance  due  at  the  be- 
ginning of  such  period,  and  this  balance  may  be  represented 
either  by  a  single  account  or  by  a  cost  account  and  an 
annulling  account. 

§  321.    Use  of  the  Reinvestment  Point  of  View 

There  are  some  cases  where,  especially  from  the  point  of 
view  of  the  debtor,  it  is  desirable  to  keep  in  view  the  entire 
original  sum.  One  of  these  cases  is  where  it  is  impossible 
or  impracticable  to  diminish  or  pay  off  the  debt  before 
maturity  and  where  accumulation  is  the  only  method  avail- 
able. Another  is  that  of  a  trust  where  there  is  an  obligation 
to  keep  the  corpus  of  the  fund  intact,  and  consequently 
reinvestment  in  some  form  is  necessary. 

But  the  calculations  of  reinvestment  are  hypothetical  and 
prospective.  They  have  not  the  same  actuality  as  those  of 
repayment,  but  are  theoretical  estimates  of  what  is  expected. 
Unless  a  contract  has  been  made  to  take  the  instalments  ofY 
one's  hands  at  a  fixed  rate,  the  amount  realized  is  pretty  sure 
to  differ  from  the  amount  anticipated. 

§  322.     Replacement 

There  is  a  third  method  of  considering  periodic  pay- 
ments, which  is  not  mentioned  in  the  actuarial  treatises,,  and 


274 


PROBLEMS  AND  STUDIES 


which  may  be  called  replacement  to  distinguish  it  from  re- 
payment and  reinvestment.  The  successive  repayments  are 
transferred  to  new  investments,  which  are  not  to  accumu- 
late but  merely  to  furnish  new  income,  helping  out  the 
diminished  income  on  the  waning  principal.  We  have  out- 
lined this  procedure  under  "Bonds  as  Trust  Fund  Invest- 
ments," in  §  148 ;  but  for  purposes  of  comparison  we  will 
put  the  materials  already  used  in  §  319  into  the  replacement 
form,  assuming  at  first  that  replacements  are  so  invested  as 
to  earn  exactly  3%. 


1 

Interest  on 
Principal 

2 

Payment  on 
Principal 

3 

Principal 
Unpaid 

4 

Replace- 
ment 

5 

Interest  on 
Replace- 
ments 

6 

Total 

Income 

1+5 

(1)  $30.00 

(2)  22.83 

(3)  15.44 

(4)  7.85 

$   239.03 
246.20 
253.59 
261.18 

$1,000.00 

760.97 

514.77 

261.18 

0. 

$   239.03 
246.20 
253.59 
261.18 

$   7.17 

14.56 
22.15 

$  30.00 

30.00 
30.00 
30.00 

$76.12 

$1,000.00 

$1,000.00 

$43.88 

$120.00 

Column  4  of  replacements  is  not  accumulative,  as  its  in- 
terest is  not  compounded,  but  is  used  as  income,  supplement- 
ing that  in  Column  1.  The  balance  of  Column  3  plus  the 
total  of  Column  4  at  any  point  make  up  $1,000.  The  two 
corresponding  amounts  in  Columns  1  and  6  always  make 
up  $30  (Column  6).  At  the  close,  the  original  $1,000  has 
been  exactly  replaced  by  the  new  securities. 

§  323.     Diminishing  Interest  Rates 

As  already  remarked,  it  would  seldom  happen  that 
exactly  3%  would  be  the  rate  secured  for  the  replacements, 
which  ought  to  be  of  the  same  grade  of  security  and  avail- 
ability as  the  original  sum.  Let  us  suppose  that  the  rate  of 
interest  was  declining  so  that  the  first  replacement  had  to 


REPAYMENT  AND  REINVESTMENT  275 

be  loaned  at  2.95%,  the  second  at  2.90%,  and  the  third  at 
2.75%.    Columns  5  and  6  are  then  the  only  ones  changed: 


5 

Interest  on 
Replacements 

6 

Total 
Income  1+5 

$  7.05 
14.19 
21.16 

$  30.00 
29.88 
29.63 
29.01 

$42.40 

$118.52 

Here  we  have  the  principal  intact,  and  the  falling-off  is  a 
gradual  one  affecting  the  interest.  If  we  had  proceeded  on 
plan  No.  (2),  the  full  predicted  interest  would  have  been 
consumed,  but  the  principal  would  have  been  impaired,  which 
is  inadmissible.  Hence,  in  cases  of  this  kind,  we  must  use 
the  vanishing  principal  with  actual  replacement.  The  re- 
investment scheme  is  a  basis  of  calculation  only  and  cannot, 
like  the  repayment  plan,  be  reduced  to  practice. 

§  324.     Proof  of  Accuracy 

It  is  interesting  to  note  that  in  the  repayment  method 
the  work  may  at  any  point  be  tested  by  a  fresh  calculation, 
showing  the  whole  procedure  to  be  coherent  and  consistent. 
For  instance,  in  our  example,  the  principal  at  three  periods 
from  maturity  is  $760.97.  Treating  this  as  the  principal, 
to  find  the  sinking  fund  we  divide  $760.97  by  2.82861,  just 
as  for  4  periods  we  divided  $1,000  by  3.7171.  This  gives 
$269.03 — the  same  result  as  before  for  the  value  of  the 
equivalent  annuity;  $22.83  as  the  interest  ($760.97  X  .03), 
and  $246.20  as  the  first  repayment  or  the  constant  reinvest- 
ment, in  either  aspect. 


2^6  PROBLEMS  AND  STUDIES 

§  325.     Varying  Rates  of  Interest 

It  must  not  be  supposed  that  there  is  at  any  one  moment 
a  single  rate  of  interest  prevaiHng.  Considerations  of  se- 
curity, convenience,  and  availabihty  give  rise  to  different 
grades  of  securities  and  different  rates  of  interest.  The 
prudent  investor  will  probably  have  at  the  same  time  some 
capital  out  at  high  rates  and  some  at  low.  The  money  at 
high  rates  is  not  quite  so  secure,  not  quite  so  readily  realiz- 
able, and  requires  more  effort  for  the  collection  of  its  in- 
come. That  at  low  rates  is  nearer  to  absolute  freedom  from 
risk  and  from  the  labor  of  supervision;  it  almost  automat- 
ically collects  its  own  income.  The  investor  will  have  so 
planned  his  investments  as  to  endeavor  to  preserve  a 
judicious  equilibrium  between  different  grades  of  security, 
and  consequently  of  income.  As  his  investments  are  liqui- 
dated, he  will  try  to  maintain  or  improve  this  equilibrium, 
and  he  will  choose  his  reinvestments  from  a  wide  range, 
some  of  low  revenue  but  highest  safety  and  others  of  the 
contrary  qualities.  It  is  therefore  fallacious  to  assume  that, 
as  an  author  has  said,  "on  the  same  day  and  under  the  same 
circumstances  money  received  from  any  one  source  may  be 
invested  at  the  same  rate  as  that  received  from  any  other 
source."  Theoretically  it  may  be,  but  practically  it  will 
usually  be  invested  in  the  same  grade  of  security  as  that 
which  it  replaces. 

§  326.    Dual  Rate  for  Income  and  Accumulation 

When  the  lender  assumes  great  risk,  or  when  the  supply 
of  loanable  capital  is  temporarily  deficient,  he  will  exact  very 
high  rates,  or  refuse  to  loan.  Or  he  may  require  a  high 
rate  and  also  demand  that  the  instalments  of  repayment  shall 
be  large  enough  to  secure  the  higher  rate  on  the  entire 
original  loan  until  fully  paid;  while  in  ordinary  reinvest- 
ments a  lower  rate  is  easily  obtained, 


REPAYMENT  AND  REINVESTMENT 


277 


§  327.     Instalment  at  Two  Rates 

Suppose  that  $1,000  is  loaned,  repayable  in  4  instalments, 
on  such  a  basis  that  the  lender  will  have  5%  interest  per 
period  on  the  entire  capital,  while  it  will  be  replaced  by 
accumulating  at  3%. 

The  sinking  fund  is  exactly  the  same  as  in  our  previous 
example,  $239.03.    But  the  instalment  is : 

not  $30 +  $239.03,  or  $269.03 
but  $50  +  $239.03,  or  $289.03 

The  instalment  here  is  as  much  greater  as  the  interest  is 
greater.  The  accumulation  is  precisely  the  same  as  hereto- 
fore. The  instalment  provides  not  only  5%  on  the  money 
remaining  invested,  but  also  2%  (unearned)  on  that  which 
had  been  repaid. 

An  instalment  of  only  $282.01  would  pay  5%  on  the 
outstanding  capital,  which  would  gradually  be  replaced  by 
3%  investments.  Thus  it  is  seen  that  the  borrower  has 
to  pay  more  than  5%  ;  in  this  instance  about  6.( 


§  328.    Amortization  of  Premiums  at  Dual  Rate 

This  loaning  at  a  dual  rate  is  of  so  little  practical  im- 
portance, at  least  in  this  country,  that  it  would  not  be  worth 
mentioning,  except  that  a  few  writers  have  tried  to  apply 
the  same  principle  to  the  amortization  of  premiums.  They 
assume  that  there  is  no  other  way  of  ascertaining  the  value 
of  a  bond  than  by  laying  aside  the  excess  of  interest  and 
letting  it  accumulate  till  maturity.  But  this  is  not  at  all 
necessary.  The  question  is,  what  uniform  rate  is  yielded 
by  each  dollar  of  the  investment  during  the  time  it  is  out- 
standing. When  this  is  ascertained,  it  can  make  no  differ- 
ence what  is  done  with  the  capital  after  it  is  returned.  We 
may  as  well  say  that  the  rate  of  a  series  of  bonds  payable 
$1,000  each  year  and  issued  at  par,  cannot  be  determined 


2^8  »         PROBLEMS  AND  STUDIES 

until  we  Know  at  what  rate  the  amounts  were  reinvested  up 
to  the  date  of  the  last  maturity.  Reinvestment  has  nothing 
to  do  with  the  yield  of  the  original  investment.  Neverthe- 
less, two  authors  have  constructed  tables  based  upon  a  dual 
rate,  one  a  rate  of  income,  the  other  a  rate  of  accumulation, 
and  they  have  taken  the  latter  at  the  arbitrary  figure  of  4%, 
irrespective  of  the  grade  of  the  bond. 

§  329.     Modified  Method  for  Valuing  Premiums 

It  is  proper  to  give  the  method  by  which  these  results 
seem  to  be  obtained,  or,  at  least,  a  method  which  will  pro- 
duce those  results. 

As  a  preliminary  we  will  consider  the  valuation  of  a 
premium  in  a  slightly  different  way  from  any  yet  given. 

We  have  seen  that  the  premium  on  $1  is  the  present 
worth  (at  the  income  rate)  of  the  difference  of  rates.  We 
may  modify  this  by  saying  that  it  is  the  difference  of  rates 
(c  —  i)  X  the  present  worth  of  an  annuity  of  $1  (P), 
which  may  be  found  in  Table  IV.*  But  to  multiply  by  P 
is  the  same  thing  as  to  divide  by  1  ^  P,  or  1/P.  Therefore, 
another  expression  for  the  premium  is  (c  —  i)^(l/F).  But 
we  found  in  §  90  that  the  rent  (1/P)  is  the  sum  of  the 
sinking  fund  (1/A)  and  the  single  interest  (i).  There- 
fore, we  still  further  modify  our  expression: 
Premium  =  (c  —  i)  -^  (i  +  1/A) 

§  330.     Rule  for  Valuation  of  a  Premium 

Rule:  Subtract  the  income  rate  from  the  cash  rate,  and 
use  this  as  a  dividend.  Add  the  instalment  from  Table  V* 
to  the  rate  of  income,  and  this  will  be  the  divisor.  The 
quotient  will  be  the  premium. 

Example :  What  is  the  premium  on  a  6%  bond  (semi- 
annual) for  $1,  50  years,  yielding  6%? 


In  Chapter  XXXII. 


REPAYMENT  AND  REINVESTMENT 


279 


c  =  .OS;  i  =  .025 ;  c  —  i=  .005  (dividend) 

1/A  at  21/2%,  100  periods  =  .002312  (Table  V*) 

.025 +  .002312  =  .027312  (divisor) 

Premium  =  .005  -^  .027312  =  .18307 

Value  of  bond,  $1.18307 

§  331.     Computation  at  Dual  Rate 

To  introduce  the  feature  of  an  accumulative  rate  differ- 
ing from  the  income  rate,  it  is  only  necessary  to  change  one 
term  in  the  above  formula.  The  value  of  1/A  must  be  taken 
from  the  column  of  Table  V,*  which  represents  the  accumu- 
lative rate,  i  remaining  as  the  income  rate. 

Example :  What  is  the  premium  on  a  6%  bond,  as  above, 
yielding  5%  on  the  entire  investment  to  maturity,  the 
principal  being  replaced  by  a  sinking  fund  at  4%  ? 

c  — i  =  .03  — .025  =  .005  (dividend) 

1/A  at  2%  ='.003203  (Table  V*) 

i  +  1/A  =  .025  +  .003203  =  .028203  (divisor) 

.005 -^  .028203  =  .17729 

Value  of  bond  =  $1.17729,  agreeing  with  Croad's 
and  Robinson's  tables. 
The  constant  income  is  .0294322  (i.e.,  2%%  of  the  value 
of  the  bond),  which  subtracted  from  the  cash  received  .03, 
leaves  as  contribution  to  the  sinking  fund  .0005678.  At  4% 
an  annuity  of  .0005678  will  amount  in  50  years  to  .17729||, 
as  may  be  ascertained  from  Table  III,*  thus  replacing  the 
premium. 

§  332.     Dual  Rate  in  Bookkeeping 

This  form  of  valuation,  which  introduces  an  arbitrary 
element,  cannot  be  satisfactorily  applied  in  the  bookkeeping 
processes  of  Chapter  XVII.  It  is  impossible  to  derive  one 
value  from  another  consistently.    The  result  will  not  agree 

*  In  Chapter  XXXII. 


228o  PROBLEMS  AND  STUDIES 

with  a  fresh  calculation,  and  the  profit  or  loss  on  a  sale  will 
be  distorted.  Any  intermediate  value,  as  shown  by  the 
actuaries,  may  have  three  different  versions. 

§  333*     Utilization  of  Dual  Principle 

While  tables  on  a  fixed  replacement  rate  are  useless  for 
purchasing  securities,  the  principle  may  occasionally  be 
utilized.  Thus,  the  trustee  referred  to  in  §  148  may  find 
that  it  is  impracticable  to  invest  favorably  such  small 
amounts  as  $400  or  $500,  and  may  conclude  to  deposit  a 
sinking  fund  in  a  savings  bank  where  he  may  reasonably 
expect  that  it  will  accumulate  for  the  next  five  years  at 
3%%,  or  he  may  make  a  contract  with  a  trust  company  on 
the  same  terms.  He  may  then  decide  also  that  it  is  better 
for  the  beneficiary  to  receive  a  uniform  income,  rather  than 
one  gradually  decreasing. 

At  13/4%  per  period,  a  sinking  fund  of  $.092375  will,  in 
ten  periods,  amount  to  $1;  therefore,  by  multiplication  it 
will  take  a  sinking  fund  of  $414,883  to  accumulate  to  $4,- 
491.29  in  10  periods.  Out  of  the  coupon  of  $2,500  must 
be  taken  the  instalment  of  $414.88,  leaving  for  the  bene- 
ficiary a  constant  semi-annual  income  of  $2,085.12,  instead 
of  the  $2,089.83  with  which  he  would  have  begun  on  the 
replacement  plan,  and  which  would  have  gradually  fallen  to 
$2,079.82  for  the  last  half-year. 

§  334.    Installation  of  Amortization  Accounts 

When  the  accounts  of  securities  have  once  been  estab- 
lished on  the  plan  of  gradual  extinction  of  premiums  and 
discounts,  it  is  not  difficult  to  take  care  of  each  new  purchase 
as  it  comes  in,  and  to  prepare  its  appropriate  schedule,  run- 
ning if  desired  all  the  way  to  the  date  of  redemption.  When, 
however,  the  accounts  have  been  previously  kept  on  the  basis 
of  par  or  of  cost,  and  it  is  desired  to  introduce  investment 
values  instead-  the  task  is  much  greater. 


REPAYMENT  AND  REINVESTMENT  281 

§  335-     Scope  of  Calculations 

It  might  be  supposed  at  first  thought  that  it  would  be 
necessary  to  start-  the  schedules  back  at  the  date  of  pur- 
chase, but  this  is  entirely  unnecessary.  For  example,  we  find 
a  5%  bond  for  $100,000  which  20  years  ago  was  bought  for 
$112,650,  and  which  has  still  10  years  to  run.  At  the  date 
of  purchase  it  must  have  had  30  years  to  run.  Turning  to 
any  table  of  5%  bonds,  30-year  column,  we  find  that  this 
was  (within  39  cents)  a  4^4%  basis.  Turning  to  the  10- 
year  column,  it  appears  that  the  value  of  a  5-year  bond  at 
4l^%  is  $106,058.46.  It  is  sufficiently  accurate  to  begin 
with  this  value,  disregarding  the  39  cents  residue,  although 
that  residue  might  be  eliminated  by  the  proportion, 
12,649.61  :  6,058.46  ::  39  :  19 

This  would  increase  the  present  value  to  $106,058.65. 

§  336.     Method  of  Procedure  when  Same  Basis  Is  Retained 

So  long  as  the  same  basis  is  preserved,  any  number  of 
intervening  years  may  be  disregarded.  The  following  pro- 
cedure may  be  recommended : 

(1)  Make  an  accurate  list  of  the  issues  held,  giving  the 
following  particulars :  dates  of  maturity;  dates  of  purchase; 
par  value  of  each  lot;  cost  of  each  lot,  being  at  the  rate  of 
$. . . .  per  $1,000  of  par;  rate  of  interest  paid,  and  the  in- 
come basis  when  ascertained.  Leave  a  column  for  valuation 
at  a  date  one  period  earlier  than  the  proposed  date  of  trans- 
formation, 

(2)  Ascertain  on  what  income  basis  each  lot  was  bought. 
This  is  done  most  easily  by  using  the  tables.  In  these  and 
the  subsequent  calculations  it  will  be  found  advantageous  to 
use  blank  books  and  entrust  nothing  to  loose  papers.  Head 
each  calculation  with  a  statement  of  the  problem  which  it 
solves.  Paper  for  these  blank  books,  ruled  with  vertical 
lineSj  everj"  third  one  of  which  is  darker  than  the  other  two, 


282  PROBLEMS  AND  STUDIES 

will  much  facilitate  the  work,  and  it  is  desirable  to  have  the 
pages  numbered  in  a  continuous  series,  for  reference. 

(3)  Find  the  value  of  each  lot  at  the  initial  date,  which 
is,  as  already  stated,  one  period  earlier  than  the  date  on 
which  the  books  are  to  be  transformed  to  investment  values. 

(4)  Where  different  lots  of  the  same  class  have  been  pur- 
chased at  various  dates  and  prices,  their  values  at  the  various 
bases  on  the  initial  date  should  be  added  together,  giving  a 
composite  value.  Ascertain  what  is  the  income  basis  for 
the  time  yet  to  run  on  this  composite  value.  This  basis  is 
the  average  basis  for  the  remaining  time  of  the  bond. 

(5)  Having  carefully  verified  all  the  initial  values  and 
the  effective  rates,  proceed  to  calculate  the  amortization  and 
accumulation  of  each  class  for  one  period,  commencing  a 
schedule  for  each.  The  resulting  values  should  be  again 
verified  with  care,  these  being  the  values  with  which  the  new 
accounts  will  begin. 

(6)  Continue  the  calculations  of  successive  values,  carry- 
ing them  into  decimals  two  places  beyond  the  cents,  ignoring 
slight  differences  in  the  last  figure.  Copy  the  results, 
rounded  to  the  nearest  cent,  into  the  schedules,  and  complete 
the  latter.  If  time  allows,  it  is  advisable  to  calculate  each 
schedule  to  maturity,  because  no  better  proof  of  the  correct- 
ness of  the  entire  chain  of  values  can  be  had  than  the  fact 
that  the  bond  reduces  exactly  to  par  at  maturity.  But  if 
time  presses,  only  a  few  of  the  values  may  be  calculated, 
but  the  last  one  should  be  verified  by  some  independent 
method.  It  is  well  in  this  case  to  leave  in  the  blank  book 
sufficient  room  to  complete  the  calculations  for  each  schedule. 
A  reference  on  the  schedule  to  the  page  of  the  blank  book 
where  the  calculation  is  made,  will  be  useful. 

(Y)  Make  such  entries  as  will  place  the  ledger  or  ledgers 
on  the  investment-value  basis. 


Part  III — Logarithms 


CHAPTER  XXIX 

FINDING  A  NUMBER  WHEN   ITS  LOGARITHM 

IS   GIVEN 

§  337'    Logarithmic  Tables 

The  meaning  and  use  of  logarithms  have  already  been 
discussed  in  a  general  way,*  and  a  simple  three-figure,  four- 
place  table  of  logarithms  given  (§43).  The  expression 
"three-figure"  refers  to  the  number  of  figures  in  each  of  the 
numbers  of  the  table,  and  the  expression  "four-place"  refers 
to  the  number  of  decimals  in  each  of  the  corresponding 
logarithms.  In  the  table  given,  for  example,  the  logarithm 
of  7.41  is  shown  to  be  .8698. 

§  338.     Discussion  of  Logarithms 

As  previously  explained,  every  logarithm  consists  of  the 
characteristic,  or  whole  number  (which  is  frequently  zero), 
and  a  decimal  fraction.  Occasionally  the  decimal  fraction 
is  zero,  as  in  the  case  of  the  logarithms  of  .01,  .1,  1,  10,  100, 
etc.  The  decimal  fractions  which  constitute  that  part  of  the 
logarithm  requiring  tabulation  are  interminate ;  that  is,  their 
values  may  be  computed  to  any  desired  number  of  decimal 


See  Chapter  III. 

283 


284  LOGARITHMS 

places  and  the  last  place  will  still  be  inexact.  Thus,  the 
logarithm  of  2  to  20  places  is  .301  029  995  663  981 195  21+. 
In  a  4-place  table,  this  would  be  rounded 

off  to 301  0 

in  a  7-place  table,  to 301  030  0 

in  a  10-place  table,  to 301  029  995  Y 

in  a  12-place  table,  to 301  029  995  664 

The  terminal  decimal  is  never  quite  accurate,  but  is  nearer 
the  true  value  than  either  the  next  greater  decimal  or  the 
next  smaller  one.  Thus,  the  logarithm  .8698  is  nearer  the 
true  logarithm  of  7.41  than  either  .8699  or  .8697. 

§339-     Standard  Tables  of  Logarithms 

The  tables  most  in  use,  like  those  of  Vega,  Chambers, 
and  Babbage,  are  of  five  figures  and  seven  places.  A  six- 
figure  table  would  have  to  contain  ten  times  as  many  loga- 
rithms as  a  five-figure  table  and,  even  though  the  number  of 
places  were  not  increased,  the  space  occupied  would  be  ten 
times  greater  than  in  the  case  of  the  five-figure  table.  In 
the  tables  above  mentioned,  two  figures  in  addition  to  the 
five  tabulated  may  be  obtained  by  interpolation. 

§  340.     United  States  Coast  Survey  Tables 

The  tables  of  the  United  States  Coast  Survey  have  five 
figures  and  ten  places.  Nine  figures  may  be  obtained  by 
simple  proportion,  but  the  tenth  is,  for  most  purposes, 
unreliable. 

It  will,  of  course,  be  understood  that  the  more  decimal 
places  given  in  the  tables,  the  more  figures  we  can  obtain 
in  the  corresponding  numbers,  but  the  number  of  figures  (in 
the  desired  number)  can  never  be  more  than  the  number  of 
places  (in  the  corresponding  logarithm).  All  of  the  fore- 
going tables  give  auxiliary  tables  of  proportionate  parts  or 
differences* 


NUMBER    FROM    LOGARITHM  285 

§  341.     Gray  and  Steinhauser  Tables 

Tables  of  24  and  20  places  have  been  published  by  Peter 
Gray  and  Anton  Steinhauser,  respectively,  but  the  plan  for 
extending  the  number  of  figures  is  quite  different  from  the 
method  of  simple  interpolation  above  referred  to.  Both  of 
these  authors  proceed  on  the  plan  of  subdividing  the  number 
into  factors,  and  adding  together  the  logarithms  of  those 
factors. 

§342     A  Twelve-Place  Table 

For  the  accurate  computation  of  problems  in  compound 
interest,  specially  designed  tables  will  be  found  in  Chapter 
XXX.  A  limit  of  twelve  figures  has  been  selected  as  the 
most  useful  for  this  purpose.  In  the  logarithms  tabulated, 
thirteen  decimal  places  are  given,  the  thirteenth  place  insur- 
ing the  accuracy  of  the  twelfth  figure  of  the  corresponding 
number,  which  would  otherwise  sometimes  be  1,  2,  or  even 
3  units  in  error,  through  the  roundings  being  preponderant 
in  one  direction  or  the  other. 

§  343.     The  "Factoring"  Method 

The  method  used  in  finding  logarithms  within  the  scope 
of  these  tables,  but  not  directly  given  in  them,  is  that  of 
factoring,  it  being  possible  to  construct  the  logarithm  of  any 
number  of  twelve  figures  or  less  (999,999,999,999  in  all) 
by  some  combination  of  the  584  logarithms  given  in  the 
table  of  factors  (§358). 

Column  A  contains  the  logarithms  of  numbers  of  two 
figures,  11  to  99,  both  inclusive,  carried  to  thirteen  places 
of  decimals. 

Column  B  contains  the  logarithms  of  four-figure  numbers 
1.001  to  1.099,  each  beginning  with  1.  and  one  zero. 

Column  C  contains  the  logarithms  of  six-figure  numbers 
J.QOOOl  to  1.00099,  each  beginning  with  1.  and  three  zeroes. 


286  LOGARITHMS 

Column  D,  1.0000001  to  1.0000099,  beginning  with  1. 
and  five  zeroes. 

Column  E,  1.000000001  to  1.000000099,  beginning 
with  1.  and  seven  zeroes. 

Column  F,  1.00000000001  to  1.00000000099,  beginning 
with  1.  and  nine  zeroes. 

For  example,  opposite  34  in  the  table  we  find  : 

A  .531  478  917  042,3  In     . ., 3.4 

B  .014  520  538  757,9  In     1.034 

C  .000  147  635  027,3  In 1.00034 

D  .000  001  476  598,7  In 1.0000034 

E  .000  000  014  766,0  In     1.000000034 

F  .000  000  000  147,7  In     . . ., 1.00000000034 

By  omitting  all  the  prefixed  zeroes,  the  printed  table  is 
made  very  compact,  each  complete  line  across  the  table  of 
factors  shown  in  §  358  containing  only  57  figures  instead  of 
82,  as  would  otherwise  be  necessary.  In  using  the  tables 
this  must  be  taken  into  consideration,  and  accordingly  it  will 
be  understood  hereafter  that  C  34,  for  example,  means  the 
number  1.00034,  and  F  34  means  1.00000000034. 

§  344.     Finding  a  Number  from  Its  Logarithm 

In  this  process  there  are  two  stages :  first,  to  divide  the 
logarithm  into  a  number  of  partial  logarithms  taken  from 
those  contained  in  the  table  of  factors;  second,  to  multiply 
together  the  numbers  corresponding  to  these  logarithms.  Of 
course  only  the  decimal  part  of  the  logarithm  is  used,  and 
the  number  has  the  position  of  its  units  figure  determined 
from  the  characteristic  of  the  logarithm. 

Let  the  logarithm  .753  797  472  366,5  be  one  which 
has  been  obtained  as  the  result  of  an  operation,  and  let  the 
corresponding  number  be  required.  Search  in  Column  A 
for  the  highest  logarithm  which  does  not  exceed  the  given 


NUMBER    FROM    LOGARITHM 


287 


one.    This  is  found  to  be  .748  188  027  006,2,  which  stands 

opposite  56. 

Subtracting  from ,. 753  797  472  366,5 

A  56     .748  188  027  006,2 


we  have  the  remainder 

This  is  smaller  than  any  logarithm 
in  Column  A.  We  search  for  it  in 
Column  B  and  find  opposite  13 
precisely  the  same  figures 


5  609  445  360,3 


6  609  445  360,3 


These  two  logarithms  added  together  make  the  given  loga- 
rithm ;  hence  the  product  of  their  numbers  gives  the  number 
required. 

To  multiply  56  by  1.013  : 


56 
1013 


56 


56 
168 


56728 


1013 

5065 
6078 

56728 


This  process  may  be  greatly  simplified  as  follows,  plac- 
ing the  figures  of  the  multiplier  in  vertical  order  at  the  side : 


56 


56 
168 


or 


56 
13  X  5     065 
13  X  6       078 


56728 


56728 


Notice  that  the  first  product  is  moved  two  columns  to  the 
right  of  the  multiplicand. 

The  column  G  used  in  the  following  example  is  not 
given  in  the  table  of  factors,  but  it  is  found  by  simply  taking 
the  first  two  figures  from  E.  The  "G"  number  in  this  case 
may  be  either  55  or  56,  which  may  make  the  thirteenth 
figure  of  the  result  doubtful,  but  probably  not  the  twelfth. 


288 


LOGARITHMS 


Now  take  a  larger  logarithm .... 
and  continue  the  subtraction         A  56 


B13 


C26 


D29 


E  58 


F48 


G55 


753  911  659  107,4 
748  188  027  006,2 


5  723 
5  609 

632 
445 

101,2 
360,3 

114 
112 

186 
901 

740,9 

888,7 

1 

1 

284 
259 

852,2 
452,2 

25 
25 

400,0 
189,1 

210,9 
208,5 

2,4 
2,4 


(See  Note  2*) 


(See  Note  1*)     2 
6 


(See  Note  3*)  5 
8 
4 

8 
5 
_5 
(See  Note  4*) 


5600 
56 
168 


567280000 
113456 
340368 


5674274928000 

* 

11348550 

5106847 


5674291383397 

•   283715 

•      45394 

•        2270 

•  454 

•  28 

3 


567429171526 


"  Qlj  following  page, 


NUMBER    FROM    LOGARITHM  289 

Note  1 :  The  second  multiplication  jumps  its  right-hand 
figure  (6)  four  places  to  the  right,  which  may  be  marked  off 
by  four  zeroes,  or  four  dots. 

Note  2 :  Having  extended  the  product  to  include  the 
13th  figure,  contraction  begins  in  this  multiplicand ;  its  first 
figure  used  being  the  7th  (marked  'A')  allowing  for  the 
carrying  from  the  8th.  Thus  the  starting  point  for  this 
multiplication  is  moved  six  places  hack. 

Note  3 :  The  multiplicand  need  no  longer  be  extended, 
as  has  been  done  at  successive  stages  above,  but  remains  the 
same  to  the  end.  For  convenience,  dots  may  be  placed  in 
advance  under  the  first  figure  to  be  used  in  multiplication  in 
each  line. 

Note  4 :  The  thirteenth  figures  are  added,  but  only  used 
for  carrying  to  the  twelfth.  In  this  example  the  total  of  the 
last  column  is  31,  but  it  does  not  appear,  except  as  con- 
tributing 3  to  the  next  column. 

The  dot  below  a  figure  indicates  where  the  contracted 
multiplication  begins,  all  the  figures  to  the  right  being 
ignored,  except  as  to  their  carrying  power. 

§  345.     Procedure  in  an  Unusual  Case 

Required  the  number  for  log.  Oil  253  170  127. 

In  this  example  there  is  no  suitable  logarithm  in  A  and 
we  must  begin  with  B,  as  shown  on  page  290. 

This  example  illustrates  the  procedure  when  B  furnishes 
the  first  logarithm.  It  also  shows  the  convenience  of  using 
paper  ruled  for  the  purpose. 

In  order  to  set  down  the  partial  products  without  hesita- 
tion, remember  the  numbers  2,  4,  6. 

In  multiplying  by  B,  the  first  figure  of  the  product  moves 
two  places  to  the  right. 

In  multiplying  by  C,  the  first  figure  of  the  product  moves 
four  places  to  the  right. 


290 


LOGARITHMS 


1 — -.1 

Formation  of  Number  from  Logarithm 

Logarithm 
A          — 

B26 

C24: 

D36 
E63 
F83 
G67 

0 

1 

1 

2 

5 

3 

1 

7 

0 

1 

2 

7 

0 

1 

1 

1 

4 

7 

3 

6 

0 

7 

7 

5 

8 

1 
1 

0 
0 

5 

4 

8 
2 

0 
1 

9 

8 

3 
1 

5 

7 

1 
0 

2 
0 

1 
1 

5 
6 

9 
6 

1 
3 

1 
4 

8 
5 

1 

7 

2 
3 

2 

2 

7 
7 

7 
3 

2 
6 

3 
0 

9 
5 

3 
3 

6 
6 

3 
0 

4 
5 

2 
2 

9 
9 

A 
B 

26 

C   2 
4 

D  3 
6 

E  6 
3 

F   8 
3 

G   6 
7 

1 
1 

0 
0^ 

2 

6 

2 

0 
4 

6 

1 

* 

2 
0 

4 

— 

— 

— 

1 

0 

2 

6 

2 

4 

• 

6 

• 
3 

2 

0 
6 

4 

7 
1 

8 
5 

7 

7 

3 

4 

9 

8 

1 

• 

0 

• 

2 

• 

6 

• 

2 

• 

4 

9 

9 

3 
6 

4 
1 
3 

4 
5 
0 

8 

8 
7 
7 
2 
3 

7 
5 
9 
1 
1 
6 
1 

1 

0 

2 

6 

2  5 

0 

0 

0 

0 

0 

0 

NUMBER    FROM    LOGARITHM 


291 


In  multiplying  by  D,  the  first  figure  of  the  mtdtipliccmd 

moves  six  places  to  the  left. 
The  following  rule  may  now  be  formulated  for  this 
process. 

§  346.    Rule  for  Finding  Number  when  Logarithm  Is  Given 

(a)  By  successive  subtractions  separate  the  given  loga- 
rithm into  a  series  of  partial  logarithms  found  in  the 
columns  of  the  table  of  factors,  setting  opposite  each  its 
letter  and  number. 

(b)  By  successive  multiplications  find  the  product  of  all 
the  numbers  thus  found,  allowing,  in  the  placing  of  the 
partial  products,  for  the  prefixed  1  and  zeroes. 

The  work  may  be  made  to  occupy  fewer  lines  by  setting 
down  the  factors  E,  F,  and  G  as  one  number  at  the  top, 
multiplying  it  by  A,  and  incorporating  it  thereafter  as  one 
multiplicand  with  the  preceding  figures.  The  result  will 
not  be  affected.  Let  the  factors  be,  as  in  the  first  example : 
A  56,  B  13,  C  26,  D  29,  E  58,  F  48,  and  G  55. 

E    F    G 

584855 


A  56  2924275 

350913 

5600000327519 

B13     56000003275 

16800000982 

5672800331776 

C26       1134560066 

340368029 

5674275259871 

D29         11348551 

5106848 

5674291715270 


292  LOGARITHMS 

Required  the  number  whose  logarithm  is  .6  or  ^. 


A  31 

.500 
491 

000  000  000,0 
361  693  834,3 

B20 

8 
8 

638  306  165,7 
600  171  761,9 

COS 

38  134  403,8 
34  742  168,9 

D78 

3  392  234,9 
3  387  483,7 

ElO 

4  751,2 
4  342,9 

F94 
G03 

408,3 
408,2 

0,1 

The  resulting  factors,  A  31,  B  20,  C  08,  D  78,  E  10, 
F  94,  and  G  03,  when  combined  produce  the  result 
3. 1622776601  7.  The  multiplication  illustrates  how 
zeroes  are  treated  when  they  occur  in  the  multipliers.  The 
result  is  the  square  root  of  10,  to  12  places,  as  may  be 
demonstrated  by  multiplying  3. 16227766017  by  itself. 

§  347.     Method  by  Multiples 

In  order  to  facilitate  the  multiplication  of  the  factors, 
A,  B,  C,  etc.,  the  table  of  multiples*  (§361),  giving  the 
product  of  each  number  from  1  to  9,  by  every  number  from 
2  to  99,  will  be  found  convenient.  Thus,  the  multiples  of 
89  read  in  one  line  as  follows : 


•Devised  by  Arthur  S,  Little. 


NUMBER    FROM    LOGARITHM  293 

123456789 

089  178  267  356  445  534  623  712  801 

Then,  if  it  be  desired,  for  example,  to  multiply  68792341 
by  89,  we  would  select  from  the  above  table 
under  6  5  3  4 

8  712 

7       623 

9  801 

2  178 

3  267 

4  356 

1  089 


6122518349 


We  have  thus  multiplied  each  figure  of  the  multiplicand 
by  both  figures  of  the  multiplier,  setting  down  each  partial 
product  unhesitatingly.  Three  figures  must  be  set  down 
for  each  partial  product,  even  if  the  first  be  a  zero.  The 
work  may  be  made  more  compact  by  piling  the  partial 
products  like  bricks,  using  only  three  lines : 

5  3  4,8  0  1,3  5  6, 
7  1  2,1  7  8,0  8  9 
6  2  3,2  6  7, 


6122518349 


To  use  this  method  in  combining  the  factors  of  a  num- 
ber, the  letters  A,  B,  C,  etc.,  are  written  above  alternate 
figure  spaces,  which  is  facilitated  by  the  use  of  paper  properly 
ruled.  Then  the  first  partial  product  under  each  letter  is 
placed  with  its  middle  figure  under  that  letter  at  the  top. 

The  following  is  an  example  of  a  combination  already 
performed  in  another  form : 


294  LOGARITHMS 

A  B  C  D  E  F  G 

A  56   1         684856 


2  8  0,4  4  8 
4  4  8,2  8  0 

2  2  4,2  8 


56       327519 

B 13    0  6  5      0  3  9,0  6 

0  7  8      0  2  6,2 

091 

6672800331778 
C26      13  0,0  5  2,0  0  0, 
1  5  6,2  0  8,0  7  8 
1  8  2,0  0  0,0  8 

5674275259864 

D  29         1  4  5,1 1  6,1  6 

1  7  4,0  5  8,1 

2  0  3,2  0  3 

567429171526 

A  process*  for  verifying  a  numerical  result,  by  using 
a  different  set  of  factors  in  a  second  operation,  is  as  follows : 

Required  the  number  corresponding  to 
.305  773  384  163,0 

The  factors  are  A  20,  B  10,  C  97,  D  21,  E  94,  F  94, 
and  G  33 ;  and  the  number  is  2.02196383809. 

In  order  to  check  the  result  and  make  sure  of  perfect 
accuracy,  we  may  solve  the  problem  a  second  time,  using  a 
smaller  factor  for  A,  provided  the  first  remainder  be  less 
than  B  99,  or  .040997692423,5.  Using  A  19  instead  of 
A  20: 


Suggested  by  Arthur  S.  Little. 


NUMBER    FROM    LOGARITHM 


295 


A  19 

305 

278 

773  384  163,0 
753  600  952,8 

B64 

27 
26 

019  783  210,2 
941  627  959,0 

C17 

78  155  251,2 
73  823  787,1 

D99 

4  331  464,1 
4  299  494,1 

E73 

31  970,0 
31  703,5 

F61 
G37 

266,5 
264,9 

1,6 

The  new  factors  are  A  19,  B  64,  C  17,  D  99,  E  73,  F  61, 
and  G  37. 

By  multiplication,  we  obtain  the  same  result  as  before : 

A  19 


B64 


C17 


D  99 


7  361,37 
6  625,233 

19 
11 

13  986,60 
4      839,2 
76      65,9 

202 

160  014  881,7 
20  216  001,5 
14  151  201,1 

202 

194  382  084,3 

1  819  749,5 

181  974,9 

202  196  383  809 


CHAPTER  XXX 

FORMING  LOGARITHMS;  TABLES 

§  348.     Explanation  of  Process 

To  form  the  logarithm  of  a  given  number — the  table  of 
factors  being  used — ^two  processes  are  necessary:  first,  the 
number  is  separated  into  a  series  of  factors  corresponding  to 
the  six  columns  of  the  thirteen-place  table ;  second,  the  loga- 
rithms of  these  factors  are  taken  from  the  table  and  added 
together. 

The  factoring  is  effected  by  a  progressive  division,  as 
illustrated  by  the  following  simple  example : 

To  find  the  logarithmic  factors.  A,  B,  C,  etc.,  of  5.6728. 
First  extend  the  number  to  12  places,  567  280  000  000. 
The  first  factor.  A,  is  always  the  first  two  figures  of  the 
number  itself. 

A  56)56  7  2  80  000  000  (1.013  B 
56 

72 
56 


168 
168 


It  will  readily  be  seen  that  one  56  might  have  been 
omitted. 

A  56)7  280  000  000  (B  13 
56 


168 
168 


296 


FORMING  LOGARITHMS 


297 


Turning  then  to  the  table,  we  have  only  to  set  down  the 
logarithms  of  these  two  factors : 

A  56    nl    748  188  027  006,2 
B  13    m/         5  609  445  360,3 


56728       nl     753  797  472  366  5 

B  13  may  be  regarded  as  an  abbreviation  of  1.013. 

In  the  next  example  a  second  divisor,  at  least,  is  required. 

A  56)  7  42  9  17152  6  (B  13 
56 


182 
168 


AB56728)14 

The  second  divisor  is  the  product  of  A  and  B.    It  might 
be  obtained  in  either  of  three  ways : 

By  multiplication  56  X  1.013  =  56728 
By  addition  56 

+       56 

+       168 


56728 
But  the  easiest  way  is 

by  subtraction       56742  (first  five  figures  of  the  number) 
—       14  (the  remainder) 


56728 


This  is  the  proper  method  for  forming  all  divisors  after 
the  first;  that  is,  subtract  the  remainder  from  the  original 
number  so  far  as  used. 

We  resume  the  division,  bringing  down  four  more 
figures,  to  the  ninth  inclusive : 


298  LOGARITHMS 

AB  )  56728   )149171626(C26 
113456 


357155 
340368 


ABC  )   5674275)   *1678726(D29 

1134855 


543871 
510685 


56742914        33186  (E 58 

28371 


4815 
4539 


276  (F  48,  7 

227 


49 
45 


The  third  divisor  A  B  C  is  also  formed  by  subtracting 
from  the  number       5674291715 
iiC  the  remainder  16  7  8  7 


leaving  5674274928 

As  only  six  figures  are  needed  for  the  divisor  and 
one  additional  figure  for  carrying,  this  is  rounded  up  to 

5  6  7  4  2  7,5 

The  fourth  divisor  is  practically  the  number  itself  so  far 
as  needed,  and  this  lasts  to  the  end. 

The  entire  process  is  now  repeated,  but  to  insure  greater 


FORMING  LOGARITHMS 


299 


accuracy  in  the  twelfth  figure  we  will  divide  out  to  the 
thirteenth  : 

A  56)  742  9  17152  6,0  (B  13 
56 

182 
168 


A  B  56  728)  149  171      (C26 
113456 


357155 
340368 


A  B  C  56  742  749  28)       16  7  8  7  2  6,0  (D  29 
(Contracted  division  begins  here)  1 1  3  4  8  5  5,0 


5  4  3  8  7  1,0 

6  1  0  6  8  4,7 


56  742  92)  3  3  18  6,3  (E  58 

2  8  3  7  1,4 


4  8  1  4,9 
4  5  3  9,4 


2  7  5,5  (F48 
2  2  7,0 


485 
454 


3  1  (G55 
28 


300  LOGARITHMS 

It  remains  only  to  add  together  the  logarithms 
A  56   (nl)    748  188  027  006,2 


B  13 

5  609  445  360,3 

C  26 

112  901  888,7 

D  29 

1259  452,2 

E  58 

25  189,1 

F  48 

208,5 

G  65 

2,4 

567  429  171526   (nl)    753  911659  107 

The  figures  in  the  thirteenth  column  are  used  only  for 
carrying  to  the  twelfth  column. 

§  349.     Rule  for  Finding  a  Logarithm 

We  may  now  formulate  the  following  rule  for  finding 
the  logarithm : 

(a)  Fix  the  number  at  13  figures,  by  adding  ciphers  or 
cutting  off  decimals. 

(b)  Cut  off  the  two  left-hand  figures  by  a  curve,  giving 
A. 

(c)  Divide  the  next  three  figures  by  A,  giving  the  two 
figures  of  B,  and  a  remainder. 

(d)  Form  the  second  divisor  A  B,  by  subtracting  the 
remainder  from  the  first  five  figures  of  the  number. 

(e)  Bring  down  four  more  figures  to  the  remainder  and 
divide  by  A  B,  giving  the  two  figures  of  C  and  a  remainder. 

(f)  Form  the  third  (and  last)  divisor  A  B  C  by  sub- 
tracting the  remainder  from  ten  figures  of  the  number. 

(g)  Divide  the  remaining  figures  by  the  third  divisor. 
As  there  are  ten  figures  in  the  divisor  and  only  eight  in  the 
dividend,  contraction  begins  immediately.  Having  obtained 
the  figures  of  D,  the  divisor  for  E,  F,  and  G  is  simply  the 
number  itself  contracted. 


FORMING  LOGARITHMS 


301 


(h)  Write  down  the  logarithms  of  A,  B,  C,  D,  E,  and 
F,  obtained  from  the  several  columns  of  the  table  of  factors ; 
also  that  of  G,  being  the  first  two  figures  of  the  correspond- 
ing E.  The  sum  will  be  the  mantissa  or  decimal  part  of  the 
logarithm  of  the  number,  the  thirteenth  decimal  place  being 
used  for  carrying  only. 

§  350.     Examples  of  Logarithmic  Computations 

It  is  advisable,  for  the  sake  of  both  convenience  and 
accuracy,  to  make  all  of  these  logarithmic  computations  on 
paper  ruled  with  at  least  thirteen  vertical  lines,  every  third 
line  being  darker  than  the  other  two.  Space  should  be  left 
on  either  side  of  these  lines  for  writing  in  the  divisors  and 
quotients,  and  for  such  other  arithmetical  work  as  may  be 
necessary.  As  a  rule,  however,  there  would  be  few,  if  any, 
additional  arithmetical  computations  which  would  have  to 
be  performed  at  the  sides. 

A  few  examples  for  practice  are  given  below  with  the 
factors  and  the  solution  : 

56Y4=A  56  B  13  C  21  D  15  E  35  F  42  G  70 
log.  5674=3.753  889  331458 
38.8586468578        =A  38  B  22  C  58  D  31  E  39  F  02  G  25 
log.         do.  =1.589  487  673  453 

3.1415926535898+=A  31  B  13  C  41  D  16  E  33  F  11  G  91 
log.         do.  =497 149  872  694 

(This  number  is  the  ratio  of  the  circumference  of  a  circle 
to  its  diameter. ) 

1.02625=B  26  C  24  D  36  E  63  F  83 
log.  do.      =  .011  253  170  127 

This  number  begins  with  an  expression  of  the  form  B 
(1.026),  hence  no  division  by  A  occurs.  1026  is  the  first 
divisor. 


302 


LOGARITHMS 

B     1026)  2  5  0  0  C  24 

2052 


4480 
4104 

B  C  102624624)  3  7  6  0  0  0,0  D  36 
3  0  Y  8  Y  3,9 


6  8  1  2  6,1 
6  1  5  7  4,8 


6  5  5  1,3  E  63 
102625)       6  1  5  7,5 


3  9  3,8 
3  0  7,9 


8  5,9  F  83 
8  2,1 


3,8 

3,1 

7     G 

B26 

Oil  147  360  775,8 

C24 

104  218  170,0 

D36 

1  563  457,3 

E63 

27  360,6 

F83 

360,5 

G70 

3,0 

70 


log.  1.02626  =  .011253  170  127* 

§  351.     Logarithms  to  Less  Than  Twelve  Places 

The  table  of  factors  may  be  cut  down  to  any  lower  num- 
ber of  places.     In  the  example  in  §  348  it  may  be  required 


*  This  result  will   be  found  also  in  the   Table  of  Interest   Ratios,   but   even 
more  extended. 


FORMING  LOGARITHMS 


303 


to  give  9  places  only,  the  tenth  being  used  for  carrying.  We 
cut  down  the  original  logarithm  to  ten  figures,  with  a  comma 
after  the  ninth,,  and  it  becomes : 


A  56 

753  911  659,1 
748  188  027,0 

B  13 

5  723  632,1 
5  609  445,4 

C  26 

114  186,7 
112  901,9 

D  29 

1  284,8 
1  259,5 

E  58 
F  24 

A 

B  1 
3 

26,3 
25,2 

1 

56 
56 
168 

C  2 
6 

567280000 
113456 
3  4  0  3  6,8 

D2 

9 

E  5 

8 
F  2 

56742749  2,8 

1 1  3  4,9 

610,7 

2  8,4 

4,6 

1 

56742917  1,4 

The  number  is  slightly  in  error  in  its  tenth  place,  but 
correct  to  the  ninth. 


304 


LOGARITHMS 


§  352.     Tables  with  More  Than  Twelve  Places 

If  a  table  of  factors  for  18  or  some  other  number  of 
places  should  hereafter  be  prepared,  the  methods  which  have 
been  explained  would  be  applicable  to  the  new  table. 

§  353-     Multiplying  Up 

Another  method  for  obtaining  the  factors  of  the  number 
in  forming  its  logarithm*  proceeds  by  multipHcatioa  instead 
of  division,  the  latter  operation  being  notably  the  more 
laborious.  The  number,  at  first  taken  as  a  decimal  less  than  1, 
is  successively  multiplied  up  to  produce  1.000,000,000,000,0 
and  these  multipliers  are  the  A,  B,  C,  D,  E,  F,  and  G,  whose 
logarithms  added  together  make  the  cologarithm,  or  loga- 
rithm of  the  reciprocal,  from  which  the  logarithm  is  easily 
obtained. 

§  354.     Process  of  Multiplying  Up 

A  is  a  number  of  two  figures,  a  little  less  than  the 
reciprocal  of  the  number,  which  will  be  called  the  sub- 
reciprocal  of  its  two  initial  figures.  A  table  of  sub-reciprocals 
is  given  in  §  360.  The  number  multiplied  by  A  will  always 
give  a  product  beginning  with  9.  B  is  always  the  arithmetical 
complement  of  the  two  figures  following  the  nine,  or  the 
remainder  obtained  by  subtracting  those  two  figures  from 
99.  Multiplication  by  B  will  usually  give  a  result  beginning 
with  999.  C  is  the  next  complement  and  gives  five  9's, 
999,99.  D  similarly  brings  999,999,9**,***,*.  No  further 
multiplication  is  necessary,  after  D  has  been  used  as  a  factor ; 
the  six  figures  in  the  places  of  the  asterisks  are  the  comple- 
ments of  E,  F,  and  G. 

To  illustrate,  let  it  be  required  to  obtain  the  logarithm 
to  the  12th  place  of  .314  159  265  359  0.  The  object  is  to 
multiply  .314  159  265  359  up  to  1.000  000  000  000  0.     The 


♦  guggested  by  Edward  S.  Thomas  of  Cincinnati, 


FORMING  LOGARITHMS 


305 


first  step  is  to  find  the  sub-reciprocal  of  31,  or  A.  Turning 
to  the  table  of  sub-reciprocals,  opposite  31  we  find  31,  by 
which  we  multiply. 


A  31 


99  —  73  =  26 
B  26  is  therefore 
the  next  multipli- 
er; dropping  the 
last  two  figures 


(99—21)  C  78 


(99—43)  D  66 


.3  141592653590 

.9  424777960770 
314159265359 

.9  738937226129  (One  nine  secured) 


194778744523 
58433623357 

.9992149594009  (Three  nines  secured) 

6994504716 
799371968 


.9999943470693  (Five  nines  secured) 

49999718 
5  9  9  9  9  6  6 


(99—47)     E  52 

(99—03)     F  96 

(100—77)     G  23 

A  31  nl 
B  26 
C  78 
D  56 
E  52 
F  96 
G  23 

colog. 
log. 


.9999999470377  (Seven  nines  secured) 

52 
96 
23 


.4913616938343 

111473607758 

3386176522 

24320423 

225833 

4169 

10 

0.502850127306 
f. 4  9  7149872694 


3o6  LOGARITHMS 

§  355'     Supplementary  Multiplication 

It  may  happen,  in  the  course  of  multiplication,  that  the 
complement  of  the  figures  following  the  9  does  not  suffice 
to  secure  two  nines  more.  In  this  case,  another  supple- 
mentary multiplication  must  take  place.  I'his  occurs  in  the 
following  example,  which  has  already  been  solved  in  §  348. 

Required  the  logarithm  of  the  number 
.567  429  171  526 

In  this  example  the  C  multiplication  also  requires  an  ad- 
ditional figure.    This  seldom  occurs. 

.567  429  171526  0 
A  17     .397  200  420  068  2 


.964  629  591594  2 

B  35 

28  938  887  747  8 

4  823  147  958  0 

.998  391  627  300  0 

B  01 

998  391  627  3 

.999  390  018  927  3 

C  60 

599  634  0114 

.999  989  652  938  7 

C  01 

9  999  896  5 

.999  999  652  835  2 

D  03 

299  999  9 

.999  999  952  835  1 

E  47,  F  16,  G  49 

47  164  9 

A  17 

230  448  921  378  3 

/B  35 
I  01 

14  940  349  792  9 

434  077  479  3 

C  60 

260  498  547  4 

01 

4  342  923  1 

D  03 

130  288  3 

E47 

20  411  8 

F  16 

69  5 

G  49 

21 

colog. 

.246  088  340  892  7 

log. 

1.753  911  659  107  3 

FORMING  LOGARITHMS 


307 


As  the  multiplication  by  B  35  brings  only  998  instead  of 
999,  we  multiply  again  by  B  01,  which  brings  it  up  to  999-h. 

In  the  next  example  there  is  a  large  defect  in  the  product 
obtained  by  multiplying  by  B  85,  which  requires  an  addi- 
tional multiplication  by  B  7. 


110  175* 

A  83 

881  400  (83,  sub-reciprocal  of  11) 
33  052  5 

B  85 

914  452  5 
73  156  200 
4  572  262  5 

B  07 

992  180  962  5 
6  945  266  737,5 

C  87 

999  126  229  237,5 

799  300  983,4 

69  938  836,0 

D45 

999  995  469  056,9 

3  999  981,9 

499  997,7 

E  30,  F  96,  G  35 

999  999  969  036,5 
30  963,5 

A  83 
B  85 
B  07 
C  87 
D45 
E  30 
F  96 
G35 

919  078  092  376,1 

35  429  738  184,5 

3  029  470  553,6 

377  671  935,8 

1  954  320,8 

13  028,8 

416,9 

1,5 

colog. 

.957  916  940  818  0 

log. 

1.042  083  059  182  0 

*The  number  no  175  was  purposely  selected,  very  slightly  in  excess  of  the 
highest  number  in  column  B,  so  as  to  produce  the  shortage  of  7. 


2o8  LOGARITHMS 

§  356.     Multiplying  Up  by  Little's  Table 

.137  128  857  423  9 


A  71  0  710  715  682  846  4   (71  being  the 
213  142  355  142  0   sub-reciprocal 

49  756  849  721  3   of  13.) 

.973  614  887  709  7 

B  26    23  415  620  818  2 

1  820  262  080  0 

078  104182  2 

.998  928  874  790  1 
B  01      998  928  874  8 


.999  927  803  664  9 
C  07       69  994  946  3 


.999  997  798  6112 

D  22 

19819818 

198  198  0 

19  815  4 

.999  999  998  606  4 

01  393  6 

EFG 

A  71 

851  258  348  719  1 

B  26 

11 147  360  775  8 

B  01 

434  077  479  3 

C  07 

30  399  549  8 

D  22 

955  446  8 

E  01 

434  3 

F  39 

169  4 

G  36 

16 

862  871 142  576  1 


.137128  857  423  9 
which  is  the  log.  of  1.371  288  574  239 


FORMING  LOGARITHMS  309 

In  the  preceding  example,  Little's  table  of  multiples 
(§  361)  is  used  in  the  multiplication.  It  will  be  found  that 
the  logarithm  when  computed  has  the  same  figures  as  the 
number  itself — a  remarkable  peculiarity  which  no  other  com- 
bination of  figures  can  possess. 

§  357-     Different  Bases 

Ten  is  the  base  of  the  logarithmic  system  which  we 
have  been  explaining;  it  is  the  most  useful  of  all  systems, 
because  ten  is  also  the  base  of  our  numerical  system.  These 
are  usually  called  common,  or  vulgar,  or  Briggsian  loga- 
rithms, but  decimal  logarithms  would  seem  a  more  appro- 
priate name. 

Any  number  might  form  the  base  of  a  system  of  loga- 
rithms, but  the  only  other  in  actual  use  is  one  known  as 
the  "natural"  system,  having  for  its  base  the  number 
2.718281828459+,  known  to  mathematicians  as  e,  which  is 
the  sum  of  the  series, 

1+  1  +  -I_  +  ___! +  1 + 

■^1^1X2^1X2X3^1X2X3X4^ 

1 

etc. 


1X2X3X4X5 
This  is  only  used  in  theoretical  inquiries,  and  is  seldom 
of  utility  to  the  accountant. 


3IO 


LOGARITHMS 


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TABLES 


311 


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cm"  no"  0"  10"  On" 

in  m  NO  NO  NO 

tN.^    ^^   "^^   I^^    "-l, 

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m"  on"  rf  00"  cm" 

Tf  l^  0   CM^  10 

10    ON    (N)    S    ON 

00   CM  t^   ,-H   10 

00   »-<   Tt  t^   0 

00"  co"  rvT  ^  no" 

CM   NO   On   CO   NO 
0  rf  00  CO  t^ 

CO   NO   ON   CM   10 

0"  Tj-"  00"  co"  rC 

CM   NO   0  in   On 

00   i-H   Tj-  ts.   0 
CO   00   CM    NO   '-H 

543,3 
977,5 
411,8 
846,1 
280,4 

0   ^   ^   CM   CM 

CO    CO    CO    Tf    -^ 

10  10   NO   NO   NO 

tv.   t^  00   00   C?N 

»-H     T-H     r-H     ,_(     l-H 

SiSia^c^ 

00  CM  iO  00  CM 
Tt   Tt   ro"  <Nf  cm" 
CO   VO   On   CM   10 
tv.   '-'   10   0   Tf 

m  00  ^  ''d-  t^ 

00    CO   t^    i-H    10 

0   co^  NO^  <>   CM^ 

00"  tC  no"  lo"  m 
CM  in  00  '-H  Tf 

0   '«*•   00   CO  t^ 

in  rs.^  0  CO  in 
■^'"  co"  co"  cm"  ^" 

t^   0   CO   NO    On 
"-H    NO   0   rl-   00 

00  0^  CO  in  IN.^ 

CM  m  t^  0  CO 

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10  On  CM  NO  0\ 
00   CM    t^    ,-H    10 
0   ^    ^   CM   CM 

CM    NO   CTn   CO   NO 
0   Tj-   00   CO   l>^ 
CO    CO    CO    •^    '^ 

0    CO   NO    0    CO 

CM  NO  0  in  On 

LO   in    NO    NO    NO 

t^  0  Tf  r^  0 

CO    00    CM    NO    '-' 
t^  t^   00   00   C3N 

»— 1     ,— 1     »-H     ,_(     »-H 

on   ON   0   0   »-i 
•-H   .-1  CM   CM   CM 

0_  tN.^  On^  00^  CM^ 

u^   00   00   CO   ^ 
0  00   NO   Tl-   ^ 

CM  On  •—<  On  CO 
in  u-T  co"  no"  tC 
0   CM    0   CO   CM 
00   -<*    0   10    0 

CO   0  CM   0  '^ 
Tf"  00"  00"  in"  00" 
t^  t^  CO  in  CM 
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m  ^  CO  CM  t^ 

0    CM    CO    rt    "^ 

tV.^  Tf    ts.^  NO    ^^ 

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00   0   CM    CO  10 
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0  Tt  00  CO  r^ 

CO   CO   CO  -^   Tj- 

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t^   t^   00   00   Ch 

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'-<"  10  t^"  On"  cm" 
On  t^   On  10   NO 
CO  t^  ir>  NO  t^ 

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in  co"  i-T  oC  rC 
0  00   On   '-•  »o 
t^  CM  CM  in  r^ 

On   CM   0   Tt  CM 
t^  rj-  CO  in  in 

00_^  in  in  in  CM^ 
00"  0"  CO  no"  no" 

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ivT  .-T  00"  ivT  co" 

■^    CO    t^    Tt    ON 

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NO    NO    -^    i-H    t^ 

00  CO  "^  .-<  CO 

Tj-    lO     t^     r-l     00 

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CO    t^   t^    CO    '!i- 

Q   rj-   ,-c   CM   00 

CO  r^  0  CO  Lo 
CM  Tf  t>.  On  »-• 
t^   T-.   10   ON   Tf 

t^  00  ON  0  0 
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00   CM   NO   »-•   ir> 

0  C3N  00  t^  m 
rf  in  t^  On  '-' 

ON    CO    t>.    1-H    NO 

CO    0   t^    Tf    0 

CO  in  NO  00  0 
0   rj-  00  CM   t^ 

NO  1-H  NO  i-H  in 

I-"    CO    -^    NO    t>. 

»-<  in  ON  CO  t^ 

0   ^   ^   ^   CM 

CM    CO   CO   T}-   Tl- 

Tt  m  in  NO  NO 

i;^  r^  t^  00  00 

S^^?)^^ 

0^  00^  0^  CM   0 
cm"  0"  On"  cm"  On" 

r^  t<  to  Tt  ON 

NO   Ch   '-'    CO   00 

tv.^  ^  0\  Ol  co_^ 
C?\"  Tj-"  On"  tvT  cm" 
»-i    CO   T-i   r^   Tf 

t^   00   CO  00   0 

CO  co_  0^  00_^  in 

0"  l^"  tC  no"  no 

in  NO  NO  '-H  CM 
CO   t^   0   NO   0 

0   t^   0\   0   CM 

CO  t^  CO  m  -^ 

co_^  vo^  tN.^  vo^  in 

in  T-T  in"  in"  00 
t^   00   CO  l^   CM 

i^  NO  ON  CO  0 

00  r>x  Ti-  T-.  t^ 
0  CO  i^  0  On 

Tf  CO  00  On  r^ 
vo  On  t^  CO  '— ' 
CM   NO   On  On   On 

Tj-    0    Tt    NO    t^ 

Tl-   0   (M    On   0 
0  m  t^  m  NO 

:-»  NO  0  in  NO 
On  in  On  in  ^N, 
ON   00   CM   Tf   NO 

CO     .-H     t^     t^     0 

in  S  00  CM  0 

0  CO  CO  00  tN. 
Tl-  t^  NO  IT)  On 
C?N   0\  CO   i-H   CO 

^    T^   On   CO   CO 
CM   NO   Tf   .-•   t^ 
t-i  to  «-•  10  •^ 

00   CM    ^    CO   Tj- 
NO   0   0   00   NO 
0    CO   CM    t^    0 

On   CO   CTn   00   CM 
m  00  -^  NO  m 
0  t>.  CM   rr   -^ 

CM   tN.   t^   ^   NO 
I-"  in  On  -^  On 
CM   t^   0   CM   »-^ 

t^   -^   ^  t^   CM 

ON    -H    CO    ■^    NO 
CO    Tt    ■«;*•    "^    Tf 

IN.  T^  ir>  00  -H 

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xn  ^n  \n  \n  *j-> 

CM   CM    CO   CO   CO 

S    S    NO    S    NO 

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VO    NO    NO    NO    NO 

CM  CM  CM   CM   CM 

0   »-i   CM   CO  Th 
CO    CO    CO    CO    cO 

to  NO  t^  00  On 
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0   -H   CM   CO   ^ 
""^   -^   "*   Tj-   Tl- 

in  NO  t^  00  On 
'^  "^  -t  Tl-  1^ 

312 


LOGARITHMS 


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NO    NO    NO 

NO    NO 

VO    NO    NO 



— 1 

TABLES 


313 


*5VOt^00O\  O'-'OJcOrJ-  tovor^COOx  Q'-'C^fOrJ-  lovOt^00O\ 

i^i>.t^t^t^  00000000  00  0000000000  a\ONONO\o\  o\o\o\o\o\ 

t>.   »-H   Tf  b>.^  »-«^  '^^  00^  '"1,  "1  °0  ^^  "1  00   ^^  "1  CA   ^.  ^  ^  ^^  ^  ^  *^  ^^  Q 

ioO"^odco  rCi-Tvoo-^"  oCcorvTcvfvo"  oiooCcood  cvfvo'-^^too 

fsJcof^coTf  rfioiovovo  vot^t^OOOO  OnOnOnOO  ^»-HC^cv>ro 

fO«^<^<^fO  COCOCOfOCO  COCOCOCOCO  COCOfOTj-Tj-  rJ-Tj-Tj-Tj-Tj- 


»-H  Tt  r>.^  o  CO  ^  o\^  *-!  "^  ^\  o  CO  vo  ch  CN  "^  00  '-H^  Tt  tN^^  o  fo  vq  0\^  c^^ 

cvf  vo  o"  in  oT  CO  tC  rg"  vo"  o"  10  oT  crT  rC  csT  ^  o'  m"  oC  crT  00  cm"  vo  o"  10 

t^QTft^o  '^t^'-H'-^oo  '-iTj-oO'-Hii->  ^cgiooorq  u-)0\cgvoON 

mO-^OOco  t>.'-tvOOrf  0\cotN.cgvo  Oioa\c<oOO  CNJVO'-'IOOn 

CvjcocooOTj-  TfiOiOVOvO  vot^t^OOOO  CJnonOvOQ  »-iT-iCVirQC^ 

cococococo  cococococo  cococococo  cOcocorj--^  rJ-rf'^Tf'^ 


"^  to  vq^  tN^  CA  p^  i-H  <M  CM  co^  Tt  10  vq^  vq^  tv.^  t^_  00^  o\  0\  c^  cq  o^  cd  '-<^  't 

vd  in   rt  CO  cvT  cm"  t-T  o"  oC  oo"  ivT  vo"  10  rf^  ro'  cm"  i-T  o"  oC  00"  cjo"  tC  \o  tn  Tf 

ChCMiOOO'-H  Tj-t^OCMiO  00'-i"^t^O  CO»oO\T-iT|-  tN.OC0VP0\ 

^\OOTfO\  COt^CMM^O  Tj-ONCOt^CM  ^O^^Onco  l^CMVOO"^ 

t^OTft^O  Ttt^T-HTfOO  ^TtOO'-'iO  OOCMIOOOCM  iOO\CMVOON 

JOO'^OOco  t^t-"VOOTf  ONCOt^CMVO  OinONCOOO  CMVO'-'inON 

CMcocorOTt-  "rf   \n  m   \0   \0  vOt^t^OOOO  OnonOsOO  »-<t-HCMCMCM 

fOrocococo  cocococoeo  c*ococococo  cocororf'^  rfrfTt^Tj- 


^_^  Lo  10^  CM_^  Tj-_^  "^  ^^  '"L  ^  "1  ^>  "^^  °°^  °°„  ^^  <^  ^^  ^  ^  °0  "^  *^>  ^.  ^>  ^ 

tC  10  o"  eg  o"  in  vo"  ltT  o'  »-'"  ctT  Tt-"  irT  co"  00"  0\  tC  cm"  CO  '-T  vo"  tvT  m"  O  '-h" 

t^Tft^iOOs  OOCO"^,— ifo  OTj-coOOOO  TfVO'^tN.VO  QOVOOOm 

tx'<^OVO'-"  vO'-hioonCM  inr^ONO'-'  CMCMCM'-'O  O\t^'^»-H00 

000000t>*t^  vOVOiOrtrh  coCM'-'^-'O  CTvOOtvVOiO  r«^CM*-HQOO 

ONCOt^'-'m  C7\cot^T-Hio  O\rot^'-"io  OOCMVOOrt  OOCMVOOCO 

voO\CMvOC?\  CMVOOncovO  OncovOOco  V00cot>.0  cot^O^tN, 


tOO\«^00(M  t^i-imO"^  OOcor^CMVO  OiO0\C000  CMVO»-HlOO\ 

CMCMcorOTt  Tj-ioiOVOO  VOt^t^OOOO  0\C7\0\00  '-'^-^CMCMCVl 

COCOCOCOCO  COCOCOCOCO  COCOCOCOCO  COCOCO'^TJ-  TfrfTj-^^ 

vq  Tf  o^  t^^  Oi  o\  CO  vq^  co^  -^^^  10^  oo_^  co_  cm_^  00^  vq^  ^  *^^  *^  "^  *"!  "^  *^^  *"!.  "^ 

'-H  o"  00"  o"  cm"  vo"  CO  o"  10"  cm"  tj-"  cm"  «o"  cm"  10  o"  00"  00'  OS  t^  vo  00"  •^  Tt  co" 

lOCOOVLOOO  OOiOt^CMO  OOmOOVOm  tJ-OOVOtI-ON  l>*Tj-tX»-<CM 

CMCOCMOOVO  -^ONt^VOCM  '-"CMOCOt^  0\mc00sC7\  r-iT-Hio»-H^ 

Tt^coOTj-  mcoOvoCM  OOiOTj-ioch  t^OOO^Ti  ONTht>^OfM 

VOt^OVOTf  ioOnVOioOO  coCMTfONt^  OniocovoCM  »-<mCMTfO\ 

TtCMt^t^-"*  t>.voCMTl-CM  r^OOiOOOOO  rft^VOr-ifO  riioVOrOVO 

OOCMiOOO^  COiOt^OOON  <:^O^O^00^^  vOTfCMOt^  ^OVOCMtX 

O^^^CV,  CMCMCMCMCM  CMCMCMCMCM  CMCMCMCM'-'  r-i^OOQ. 

"^OOCMVOO  -"ii-OOCMvoo  -"^OOCMVOO  rf-QOCMVOO  rtOOCMVoSv 

^-H^CMCMco  rocOTfrtin  miovovor^  I^t^OOOOON  0\0\000 

COCOCOCOCO  rOCOCOCOCO  COCOCOCOCO  COCOCOtOCO  COCO"^^Tf 

t>._  oo_^  10  "T,  ■^^  o\^  vq  tN.^  '-H  o\^  CO  vo^  vq  CM^  o^  co^  '-j,  vq  ov  i^  00  vo  cm  10  10 

T-T  o"  cm"  o  o"  i-h"  00"  CO  vo"  r-T  Tj^  ro"  00"  o"  T}-"  oT  ^"  Lo"  co"  oC  ocJ"  os  vo"  cm"  tC 

ONOOt^ONON  Ovt^OOt^vO  r-HTj-^LOrt  coCM-^mOV  (iOcOVOONOV 

coCM'-hvocm  0\OOCOcoO  ^s.CMVO'-lVO  Tj-cocoioiO  CMOCMVOm 

COCM10CM3-H  VOOOCMCMVO  iOt-hCMCMVO  OvCMt^OOfO  lOCOThiOrf 

VOC3VCMOO\  OO'-HiOOvOO  (Mmiol^Q  OONCMrfiO  OcocoIn^Ov 

CMiOI>^vOO  C^OOOOCM  On-^CMVOO  mroOOONCX)  VOCMt^O«-H 

'-iCOO^r^  OvincoOOOS  OOOOO^CMO  CM'-<r^<Mt^  CO'-h.-hVOio 

>S;3;2iSf>J  9S°2;3Ji:5J^:  !r2^'-'00<^  ^t-rtoooocM  r<it^»N.CMco 

OOO^tOvo  O-^OOOCM  T^TJ•^or^co  CMOt^'^'-H  tN.CMt^CMVO 

IPQ^^i^  C000co0\rf  Ov'^CAThON  rhCAcooOco  tvCMVO'-fiO 

f^5959Sj^  S?Q!Z''~'^  cMcocoTf-*^  miovovoi^  t^ooooo\ON 

0000000000  0^0\o^o^o^  onO\c?no\o\  o\  a\  o\  os  a\  0\  o\  c\  0\  0\ 

lOVOr^OOON  Ot-iCMcotT  mvot^OOCTv  Or-iCMro'^  lOVOt^COOv 

t>^t^t^t>.t^  0000000000  0000000000  o\  o\  o\  o\  o\  0\  0\  0\  0\  0\ 


314 
§359. 


LOGARITHMS 


Table  of  Interest  Ratios 


1  +  i 

Logarithm 

1  +  i 

Logarithm 

1.00125 

000  542  529  092  294 

1.01 

004  321  373  782  643 

1.0015 

000  650  953  629  595 

1.01025 

004  428  859  114  686 

1.00175 

000  759  351  104  737 

1.0105 

004  536  317  851  323 

1.002 

000  867  721  531  227 

1.01075 

004  643  750  005  712 

1.00225 

000  976  064  922  559 

1.011 

004  751  155  591  001 

1.0025 

001  084  381  292  220 

1.01125 

004  858  534  620  329 

1.00275 

001  192  670  653  684 

1.0115 

004  965  887  106  823 

1.003 

001  300  933  020  418 

1.01175 

005  073  213  063  604 

1.00325 

001  409  168  405  876 

1.012 

005  180  512  503  780 

1.0035 

001  517  376  823  504 

1.01225 

005  287  785  440  451 

1.00375 

001  625  558  286  737 

1.0125 

005  395  031  886  706 

1.004 

001  733  712  809  001 

1.01275 

005  502  251  855  626 

1.00425 

001  841  840  403  709 

1.013 

005  609  445  360  280 

1.0045 

001  949  941  084  268 

1.01325 

005  716  612  413  731 

1.00475 

002  058  014  864  072 

1.0135 

005  823  753  029  028 

1.005 

002  166  061  756  508 

1.01375 

005  930  867  219  212 

1.00525 

002  274  081  774  949 

1.014 

006  037  954  997  317 

1.0055 

002  382  074  932  761 

1.01425 

006  145  016  376  364 

1.00575 

002  490  041  243  299 

1.0145 

006  252  051  369  365 

1.006 

002  597  980  719  909 

1.01475 

006  359  059  989  323 

1.00625 

002  705  893  375  925 

1.015 

006  466  042  249  232 

1.0065 

002  813  779  224  673 

1.01525 

006  572  998  162  075 

1.00675 

002  921  638  279  469 

1.0155 

006  679  927  740  826 

1.007 

003  029  470  553  618 

1.01575 

006  786  830  998  449 

1.00725 

003  137  276  060  415 

1.016 

006  893  707  947  900 

1.0075 

003  245  054  813  147 

1.01625 

007  000  558  602  125 

1.00775 

003  352  806  825  089 

1.0165 

007  107  382  974  057 

1.008 

003  460  532  109  506 

1.01675 

007  214  181  076  625 

1.00825 

003  568  230  679  656 

1.017 

007  320  952  922  745 

1.0085 

003  675  902  548  784 

1.01725 

007  427  698  525  323 

1.00875 

003  783  547  730  127 

1.0175 

007  534  417  897  258 

1.009 

003  891  166  236  911 

1.01775 

007  641  111  051  437 

1.00925 

003  998  758  082  352 

1.018 

007  747  778  000  740 

1.0095 

004  106  323  279  658 

1.01825 

007  854  418  758  035 

1.00975 

004  213  861  842  026 

1.0185 

007  961  033  336  183 

TABLES 


315 


Table  of  Interest  Ratios — {Continued) 


1  +  i 

Logarithm 

1  1  +  * 

Logarithm 

1.01875 

008  067  621  748  033 

1.0275 

Oil  781  830  548  107 

1.019 

008  174  184  006  426 

1.02775 

Oil  887  485  452  387 

1.01925 

008  280  720  124  194 

1.028 

Oil  993  114  659  257 

1.0195 

008  387  230  114  159 

1.02825 

012  098  718  181  213 

1.01975 

008  493  713  989  132 

1.0285 

012  204  296  030  743 

1.02 

008  600  171  761  918 

1.02875 

012  309  848  220  326 

1.02025 

008  706  603  445  309 

1.029 

012  415  374  762  433 

1.0205 

008  813  009  052  089 

1.02925 

012  520  875  669  524 

1.02075 

008  919  388  595  035 

1.0295 

012  626  350  954  050 

1.021 

009  025  742  086  910 

1.02975 

012  731  800  628  455 

1.02125 

009  132  069  540  472 

1.03 

012  837  224  705  172 

1.0215 

009  238  370  968  466 

1.0305 

013  047  996  115  232 

1.02175 

009  344  646  383  631 

1.031 

013  258  665  283  517 

1.022 

009  450  895  798  694 

1.0315 

013  469  232  309  170 

1.02225 

009  557  119  226  374 

1.032 

013  679  697  291  193 

1.0225 

009  663  316  679  379 

1.0325 

013  890  060  328  439 

1.02275 

009  769  488  170  411 

1.033 

014  100  321  519  621 

1.023 

009  875  633  712  160 

1.0335 

014  310  480  963  307 

1.02325 

009  981  753  317  307 

1.034 

014  520  538  757  924 

1.0235 

010  087  846  998  524 

1.0345 

014  730  495  001  753 

1.02375 

010  193  914  768  475 

1.035 

014  940  349  792  937 

1.024 

010  299  956  639  812 

1.0355 

015  150  103  229  471 

1.02425 

010  405  972  625  180 

1.036 

015  359  755  409  214 

1.0245 

010  511  962  737  214 

1.0375 

015  988  105  384  130 

1.02475 

010  617  926  988  539 

1.038 

016  197  353  512  439 

1.025 

010  723  865  391  773 

1.039 

016  615  547  557  177 

1.02525 

010  829  777  959  522 

1.04 

017  033  339  298  780 

1.0255 

010  935  664  704  385 

1.041 

017  450  729  510  536 

1.02575 

Oil  041  525  638  950 

1.0425 

018  076  063  645  795 

1.026 

Oil  147  360  775  797 

1.043 

018  284  308  426  531 

1.02625 

Oil  253  170  127  497 

1.044 

018  700  498  666  243 

1.0265 

Oil  358  953  706  611 

1.045 

019  116  290  447  073 

1.02675 

Oil  464  711  525  690 

1.046 

019  531  684  531  255 

1.027 

Oil  570  443  597  278 

1.0475 

020  154  031  638  333 

1.02725 

Oil  676   149  933  909 

1.048 

020  361  282  647  708 

3i6  LOGARITHMS 

Table  of  Interest  Ratios — (Concluded) 


1  +  i 

Logarithm 

1  +  i 

Logarithm 

1.049 

1.05 

1.055 

1.06 

1.065 

020  775  488  193  558 

021  189  299  069  938 
023  252  459  633  711 
025  305  865  264  770 
027  349  607  774  757 

1.07 

1.075 

1.08 

1.09 

1.10 

029  383  777  685  210 
031  408  464  251  624 
033  423  755  486  950 
037  426  497  940  624 
041  392  685  158  225 

§360. 


Table  of  Sub-Reciprocals 


Initial 

Sub- 

Initial 

Sub- 

Figures 

Reciprocal 

Figures 

Reciprocal 

10 

90 

35-36 

27 

11 

83 

37 

26 

12 

76 

38-39 

25 

13 

71 

40 

24 

14 

66 

41-42 

23 

IS 

62 

43-44 

22 

16 

58 

45-46 

21 

17 

55 

47-49 

20 

18 

52 

50-51 

19 

19 

50 

52-54 

18 

20 

47 

55-57 

17 

21 

45 

58-61 

16 

22 

43 

62-65 

15 

23 

41 

66-70 

14 

24 

40 

71-75 

13 

25 

38 

76-82 

12 

26 

27 

83-89 

11 

27 

35 

90 

10 

28 

34 

29 

33 

30 

32 

31 

31 

32 

30 

33 

29 

34 

28 

FORMING  LOGARITHMS 


317 


§361. 


Table  of  Multiples 

1 

2 

3 

4 

5 

6 

7 

8 

9 

001 

002 

003 

004 

005 

006 

007 

008 

009 

002 

004 

006 

008 

010 

012 

014 

016 

018 

003 

006 

009 

012 

015 

018 

021 

024 

027 

004 

008 

012 

016 

020 

024 

028 

032 

036 

005 

010 

015 

020 

025 

030 

035 

040 

045 

006 

012 

018 

024 

030 

036 

042 

048 

054 

007 

014 

021 

028 

035 

042 

049 

056 

063 

008 

016 

024 

032 

040 

048 

056 

064 

072 

009 

018 

027 

036 

045 

054 

063 

072 

081 

010 

020 

030 

040 

050 

060 

070 

080 

090 

Oil 

022 

033 

044 

055 

066 

077 

088 

099 

012 

024 

036 

048 

060 

072 

084 

096 

108 

013 

026 

039 

052 

065 

078 

091 

104 

117 

014 

028 

042 

056 

070 

084 

098 

112 

126 

015 

030 

045 

060 

075 

090 

105 

120 

135 

016 

032 

048 

064 

080 

096 

112 

128 

144 

017 

034 

051 

068 

085 

102 

119 

136 

153 

018 

036 

054 

072 

090 

108 

126 

144 

162 

019 

038 

057 

076 

095 

114 

133 

152 

171 

020 

040 

060 

080 

100 

120 

140 

160 

180 

021 

042 

063 

084 

105 

126 

147 

168 

189 

022 

044 

066 

088 

110 

132 

154 

176 

198 

023 

046 

069 

092 

115 

138 

161 

184 

207 

024 

048 

072 

096 

120 

144 

168 

192 

216 

025 

050 

075 

100 

125 

150 

175 

200 

225 

026 

052 

078 

104 

130 

156 

182 

208 

234 

027 

054 

081 

108 

135 

162 

189 

216 

243 

028 

056 

084 

112 

140 

168 

196 

224 

252 

029 

058 

087 

116 

145 

174 

203 

232 

261 

030 

060 

090 

120 

150 

180 

210 

240 

270 

031 

062 

093 

124 

155 

186 

217 

248 

279 

032 

064 

096 

128 

160 

192 

224 

256 

288 

033 

066 

099 

132 

165 

198 

231 

264 

297 

034 

068 

102 

136 

170 

204 

238 

272 

306 

3i8 


LOGARITHMS 
Table  of  Multiples — (Continued) 


1 

2 

3 

4 

5 

6 

7 

8 

9 

035 

070 

105 

140 

175 

210 

245 

280 

315 

036 

072 

108 

144 

180 

216 

252 

288 

324 

037 

074 

111 

148 

185 

222 

259 

296 

333 

038 

076 

114 

152 

190 

228 

266 

304 

342 

039 

078 

117 

156 

195 

234 

273 

312 

351 

040 

080 

120 

160 

200 

240 

280 

320 

360 

041 

082 

123 

164 

205 

246 

287 

328 

369 

042 

084 

126 

168 

210 

252 

294 

336 

378 

043 

086 

129 

172 

215 

258 

301 

344 

387 

044 

088 

132 

176 

220 

264 

308 

352 

396 

045 

090 

135 

180 

225 

270 

315 

360 

405 

046 

092 

138 

184 

230 

276 

322 

368 

414 

047 

094 

141 

188 

235 

282 

329 

376 

423 

048 

096 

144 

192 

240 

288 

336 

384 

432 

049 

098 

147 

196 

245 

294 

343 

392 

441 

050 

100 

150 

200 

250 

300 

350 

400 

450 

051 

102 

153 

204 

255 

306 

357 

408 

459 

052 

104 

156 

208 

260 

312 

364 

416 

468 

053 

106 

159 

212 

265 

318 

371 

424 

477 

054 

108 

162 

216 

270 

324 

378 

432 

486 

055 

110 

165 

220 

275 

330 

385 

440 

495 

056 

112 

168 

224 

280 

336 

392 

448 

504 

057 

114 

171 

228 

285 

342 

399 

456 

513 

058 

116 

174 

232 

290 

348 

406 

464 

522 

059 

118 

177 

236 

295 

354 

413 

472 

531 

060 

120 

180 

240 

300 

360 

420 

480 

540 

061 

122 

183 

244 

305 

266 

427 

488 

549 

062 

124 

186 

248 

310 

372 

434 

496 

558 

063 

126 

189 

252 

315 

378 

441 

504 

567 

064 

128 

192 

256 

320 

384 

448 

512 

576 

065 

130 

195 

260 

325 

390 

455 

520 

585 

066 

132 

198 

264 

330 

396 

462 

528 

594 

067 

134 

201 

268 

335 

402 

469 

536 

603 

068 

136 

204 

272 

340 

408 

476 

544 

612 

069 

138 

207 

276 

345 

414 

483 

552 

621 

FORMING  LOGARITHMS 


319 


Table  of  Multiples — (Concluded) 


1 

2 

3 

4 

5 

6 

7 

8 

9 

070 
071 
072 
073 
074 

140 
142 
144 
146 
148 

210 
213 
216 
219 
222 

280 
284 
288 
292 
296 

350 
355 
360 
365 
370 

420 
426 
432 
438 
444 

490 

497 
504 
511 
518 

560 
568 
576 
584 
592 

630 
639 
648 

657 
666 

075 
076 
077 
078 
079 

150 
152 
154 
156 
158 

225 
228 
231 
234 
237 

300 
304 
308 
312 
316 

375 
380 
385 
390 
395 

450 
456 
462 
468 
474 

525 
532 
539 
546 
553 

600 
608 
616 
624 
632 

675 
684 
693 
702 
711 

080 
081 
082 
083 
084 

160 
162 
164 
166 
168 

240 
243 
246 
249 
252 

320 
324 
328 
332 
336 

400 
405 
410 
415 
420 

480 
486 
492 
498 
504 

560 
567 
574 
581 
588 

640 
648 
656 
664 
672 

720 
729 
738 
747 
756 

085 
086 
087 
088 
089 

170 
172 
174 
176 
178 

255 
258 
261 
264 
267 

340 
344 
348 
352 
356 

425 
430 
435 
440 
445 

510 
516 
522 
528 
534 

595 
602 
609 
616 
623 

680 
688 
696 
704 
712 

765 
774 
783 
792 
801 

090 
091 
092 
093 
094 

180 
182 
184 
186 
188 

270 
273 
276 
279 
282 

360 
364 
368 
372 
376 

450 
455 
460 
465 
470 

540 
546 

552 
558 
564 

630 
637 
644 
651 
658 

720 
728 
736 
744 
752 

810 
819 
828 
837 
846 

095 
096 
097 
098 
099 

190 
192 
194 
196 
198 

285 
288 
291 
294 
297 

380 
384 
388 
392 
396 

475 
480 
485 
490 
495 

570 
576 
582 
588 
594 

665 
672 
679 
686 
693 

760 
768 
776 
784 
792 

855 
864 
873 
882 
891 

Part  IV— Tables 


CHAPTER  XXXI 

EXPLANATION  OF  TABLES  USED 

§  362.     Object  of  the  Tables 

Any  value  shown  in  the  following  tables*  might  have 
been  ascertained  by  the  rules  given  in  the  text;  but  it  is 
convenient  and  time  saving  to  have  at  hand,  already  worked 
out,  those  results  which  are  most  frequently  needed. 

§  363.     Degree  of  Accuracy 

The  tables  shown  give  each  value  to  eight  decimal  places, 
while  the  ordinary  tables  extend  only  to  five  or  six  decimals. 
This  allows  accurate  computations  to  be  made  on  sums  up 
to  one  million  dollars,  to  the  nearest  cent — a  degree  of 
accuracy  which  will  meet  any  ordinary  requirements. 

§  364.     Rates  and  Periods 

The  rates  used  in  the  tables*  are  as  follows :  1%,  1%%, 
11/2%,  1%%,  2%,  21/4%,  21/2%,  2%%,  3%,  31/2%,  4%, 
4%%,  5%,  and  6%.  These  are  the  rates  most  commonly 
used,  since  most  investments  are  on  a  semi-annual  basis. 
Rules  for  intermediate  rates  will  be  found  in  §§  375,  376. 

The  periods  given  are  from  1  to  50,  inclusive,  and  also 


•  See  Chapter  XX2CII, 

32Q 


EXPLANATION  OF  TABLES  USED       321 

every  5th  period  thereafter,  viz.:  55,  60,  65,  YO,  75,  80,  85, 
90,  95,  100.  Rules  for  obtaining  the  values  for  periods  in- 
tervening above  50,  and  for  extending  above  100  periods, 
will  be  found  in  §  374. 

In  all  the  following  tables  of  compound  interest,  the 
principal  is  considered  to  be  $1;  for  any  other  principal 
the  tabular  result  must  be  multiplied  by  the  number  of  dol- 
lars in  the  principal. 

§  365.    Tables  Shown 

Tables  for  obtaining  the  following  results  are  shown  in 
Chapter  XXXII. 

Table     I — Amount 

II— Present  Worth 
III — Amount  of  Annuity 
IV — Present  Worth  of  Annuity 

V — Sinking  Fund 
VI — Reciprocals  and  Square  Roots 

§  366.     Annuities — When  Payable 

"Annuity"  in  these  tables  signifies  the  ordinary  annuity 
where  the  payment  is  made  at  the  end  of  each  period.  This 
kind  of  annuity  is  the  one  most  used  in  investment  calcula- 
tions. Annuities  paid  in  advance,  like  the  premiums  in  life 
insurance,  are  sometimes  called  annuities  due  (§75).  Their 
amounts  and  present  worths  may  be  derived  from  the  tables 
of  ordinary  annuities. 

§  367.     Table  I — Amount 

This  gives  the  amount  to  which  $1,  invested  now  at  the 
rate  i,  will  have  accumulated  at  the  end  of  n  periods.  The 
rates  (i)  are  at  the  top  of  the  table  and  the  numbers  of 
periods  (w)  are  on  the  left-hand  margin.  Each  term  up  to 
50  periods  is  l-\-  i  times  the  term  above  it ;  or  is  the  term 


322 


TABLES 


below  it  divided  by  1  +  i.  Each  term  may  also  be  considered 
as  that  power  oil  +  i,  the  ratio  of  increase,  whose  exponent 
is  on  the  left.  Thus  in  the  column  3%,  where  i  =  .03,  the 
value  for  9  periods  is  the  9th  power  of  1.03,  or  1.03®  = 
1.30477318.  If  this  be  multiplied  by  1.03  it  gives  the  10th 
term,  1.34391638;  if  divided  by  1.03  it  gives  the  8th  term, 
1.26677008. 

§  368.     Compound  Interest 

To  find  the  compound  interest,  subtract  1  from  the 
amount.  Thus  the  compound  interest  at  3%  for  9  periods  is 
.30477318;  for  25  periods,  1.09377793.  It  is  unnecessary, 
therefore,  to  give  a  separate  table  of  the  compound  interest. 
All  the  other  tables  might  be  derived  from  Table  I.  The 
second  line  in  each  column  is  the  ratio  of  increase. 

§  369.    Table  II— Present  Worth 

This  gives  the  present  worth  of  $1  payable  n  periods 
from  now  at  the  rate  i ;  or  the  principal  which  invested  now 
will  at  the  end  of  n  periods  have  accumulated  to  $1 ;  or  $1 
discounted  for  n  periods  at  i.  It  proceeds  in  exactly  the 
contrary  manner  to  Table  I,  diminishing  instead  of  increas- 
ing, each  term  being  divided  by  l-\-  i  to  produce  the  succeed- 
ing one,  or  multiplied  by  1  +  i  to  produce  the  preceding. 

Each  term  may  be  obtained  independently  from  the  cor- 
responding term  in  Table  I,  the  two  terms  being  reciprocals 
of  each  other.  If  we  represent  any  term  in  Table  I  by  (I) 
and  the  corresponding  term  in  Table  II  by  (II),  then  we 
may  say:  (II)  =1-^- (I)  and  (I)  =1^(11);  or  more 
briefly  with  negative  exponents,    (II)  =  (I)~^  and  (I)  == 

(II)-'. 

The  second  line  in  each  column  is  the  discount  ratio  or 
reciprocal  of  1  +  f ;  and  each  term  below  is  the  nth  power  of 


EXPLANATION  OF  TABLES  USED       323 

that  number.  Thus  in  the  3%  column  the  discount  ratio  is 
1.03-^  or  .97087379;  the  present  worth  for  9  periods  is 
.76641673,  or  the  9th  power  of  .97087379,  which  may  be 
expressed  1.03"^  or  1  -^  1.03^  Multiplied  by  1.03,  it  gives 
1.03-^  or  .78940923;  divided  by  1.03  it  gives  1.03-"''  or 
.74409391.  Each  of  these  multiplied  by  its  correlative  in 
Table  I  will  give  unity:  (II)  X  (I)  =  1. 

All  these  relations  should  be  verified  by  experiment  until 
thoroughly  understood. 

The  compound  discount  is  obtained  by  subtracting  the 
present  worth  from  1.  The  compound  discount  for  9  periods 
at  3%  is  1  —  .76641673  =  .23358327.  Since  this  operation 
is  easy,  it  is  unnecessary  to  give  a  separate  table  of  compound 
discounts. 

§  370.    Table  III — Amount  of  Annuity 

This  table  gives  the  amount  to  which  an  ordinary  annuity 
will  accumulate ;  that  is,  if  $1  be  invested  at  the  end  of  each 
period,  the  total  investment  will,  after  n  periods,  reach  this 
amount. 

It  is  formed  from  Table  I  by  adding  together  the  same 
number  of  terms  as  of  the  periods  required.  The  three  top 
lines  of  Table  I  give  the  third  line  of  Table  III ;  the  fourth 
line  of  Table  III  is  the  sum  of  the  first  four  lines  of  Table  I ; 
but  for  this  purpose  the  line  marked  0  in  Table  I  must  be 
counted  in.  Thus  in  the  2%  column,  1  +  1.02  +  1.0404  in 
Table  I  gives  the  value  for  3  periods  in  Table  III,  3.0604; 
for  4  periods  it  is  1  +  1.02  +  1.0404  +  1.061208  = 
4.121608;  etc. 

According  to  the  principles  laid  down  in  §  60,  we  might 
have  proceeded  in  this  way:  Take  the  amount  of  $1  for  3 
periods  from  Table  I  (not  the  third  but  the  fourth  line), 
1.061208 ;  drop  the  1,  giving  .061208 ;  divide  by  .02,^produc- 
ing  3.0604. 


324  TABLES 

But  where  the  figures  have  been  rounded,  this  procedure 
would  leave  two  places  indeterminate.  For  example,  the 
amount  of  $1  for  20  periods  is  1.48594740;  the  compound 
interest  is  .48594740;  this  multipHed  by  100  and  divided  by 
2,  gives  24.297370.  We  have,  therefore,  cut  down  our 
result  from  8  decimals  to  6.  But  by  addition  of  the  first  20 
lines  of  Table  I,  we  get  in  Table  III,  24.29736980;  a  gain 
in  accuracy  of  two  places. 

It  will  be  better,  therefore,  to  reverse  the  process  and  test 
the  accuracy  of  the  table  by  dividing  the  result  in  Table  III 
by  100  and  multiplying  by  2. 

24.29736980  X  2  -^  100  =  .4859473960 
.4859473960  +  1  =  1.4859473960  =  1.4859474|  | 

This  not  only  tests  the  accuracy  of  the  table,  but  adds  two 
places. 

This  suggests  that  for  very  accurate  and  extensive  com- 
putations we  may  extend  Table  I  to  10  figures,  the  last  of 
which  will  be  nearly  accurate. 

For  most  questions  of  investment  6  decimals  of  Tables 
III  and  IV  will  be  ample. 

When  the  amount  of  an  annuity  due  is  required,  it  is 
obtained  as  follows  :  subtract  one  from  the  number  of  periods 
and  subtract  one  from  the  number  of  dollars.  Thus  we  have 
at  2%  : 

Amount  of  ordinary  annuity,    4  periods,  4.121608 

Amount  of  annuity  due,  3  periods,  3.121608 

§  371.     Table  IV — Present  Worth  of  Annuity 

This  table  gives  the  present  worth  of  an  annuity  and  is 
derived  from  Table  II,  precisely  as  III  is  derived  from  I. 
In  the  same  way  as  before.  Table  II  may  be  extended  two 
places ;  but  after  multiplying  by  the  rate,  the  result  must  be 
subtracted  from  1. 


EXPLANATION  OF  TABLES  USED       325 

The  present  worth  of  an  annuity  of  20  periods  at  2% 
is,  by  Table  IV,  16.35143334,  which  X  .02  =  .3270286668; 
1—. 3270286668=. 6729713332.  Table  II  gives  .67297133, 
which  is  correct  as  far  as  it  goes. 

Table  IV  is  the  one  used  in  bond  valuations  for  ascer- 
taining premiums  and  discounts  which,  as  we  have  seen, 
are  merely  present  worths  of  annuities  consisting  of  the 
difference  between  the  cash  and  income  rates.  Any  ordinary 
premium  or  discount  where  the  principal  does  not  exceed 
one  million  dollars  may  safely  be  computed  by  using  6  or 
7  figures  of  the  decimals. 

To  transform  Table  IV  of  ordinary  annuities  into  one 
of  annuities  due,  add  one  to  the  number  of  periods  and  add 
one  dollar  to  the  value. 

§  372.     Table  V— Sinking  Fund 

This  table  gives  sinking  funds.  It  answers  the  question : 
What  sum  shall  be  invested  at  the  end  of  each  of  n  periods  so 
that  the  sum-total  with  all  accumulations  shall  amount  to  $1 
at  the  end  of  n  periods  ? 

Each  term  of  Table  V  is  the  reciprocal  of  the  corre- 
sponding term  in  Table  III.  (V)  =  (III)-\  Thus,  to  find 
the  sinking  fund  necessary  to  provide  a  total  of  $1  in  9 
periods,  we  divide  1  by  the  total  to  which  an  annuity  of  $1 
would  accumulate  in  9  periods.  At  3%,  the  latter  would 
be: 

10.15910613 ;  1  -^  10.15910613  =  .0984338570 

which   is  the   sinking   fund   required,   carried   two  places 
further  than  in  Table  V. 

Another  method  of  deriving  the  sinking  fund  would  be 
to  divide  the  single  interest  (.03)  by  the  compound  interest 
(.30477318)  of  $1  from  Table  I,  which  will  be  found  to 
give  the  same  result :  i-^l. 


326  TABLES 

§  373'     Rent  of  Annuity 

To  find  what  annuity  has  a  present  worth  of  $1,  we 
have  only  to  add  to  the  rate  of  interest  the  sum  taken  from 
Table  V.  This  gives  the  rent  of  an  annuity  which  $1  will 
purchase,  and  it  is,  therefore,  unnecessary  to  provide  a 
table  for  that  purpose.  It  might  also  be  obtained  to  8 
places  from  Table  II,  dividing  the  single  interest  by  th? 
compound  discount.  It  could  also  be  derived  by  finding 
the  reciprocal  of  the  corresponding  term  in  Table  IV. 

§  374.     Extension  of  Time 

The  tables  go  as  far  as  100  periods  only,  but  Tables  I 
and  II  may  be  extended  to  as  many  periods  as  desired  by 
multiplication.  The  values  for  148  periods  might  be  ob- 
tained by  multiplying  together  those  for  100  and  48  periods 
respectively.     Thus,  at  1%,  Table  I,  we  have: 


100 

2.70481383 

48 

1.61222608 

Using  contracted  multiplication, 

2.70481383 

1.62288830 

2704814 

540963 

64096 

5410 

1623 

22 

4.36077141 

The  last  figure  is  not  quite  accurate,  but  we  could  have 
made  it  more  so  by  getting  10  figure  values  for  100  and  for 
48  periods  from  Table  III. 


EXPLANATION  OF  TABLES  USED 


327 


100(170.48138294X.01=1.7048138294)+1=2.7048138294 
48 (  61.22260777X01=  .6122260777)4-1=1.6122260777 


2.7048138294 

1.6228882976 

270481383 

64096277 

5409628 

540963 

162289 

1893 

189 

19 


Correct  result  to  10  figures,       4.3607713911 

To  extend  Table  III,  IV,  or  V  as  to  time,  it  is  easiest 
to  extend  Table  I  or  II  and  thence  derive  the  value  required. 

§  375*    Subdivision  of  Rates 

Although  the  rates  given  in  these  tables  are  those  most 
frequently  required,  yet  it  often  happens  that  intermediate 
rates  occur,  especially  in  bond  computations.  It  might  be 
supposed  that  these  inter-rates  could  be  obtained  by  "split- 
ting the  difference"  into  as  many  parts  as  necessary.  But 
a  trial  will  show  that  this  gives  only  a  rough  approximation. 

In  Table  I,  for  10  periods  at  the  rate  3%,  the  amount 

is   1.34391638 

and  at  2%%  it  is 1.28008454 


Midway  between  them  is 1.31200046 

but  this  is  not  the  true  value  for  23^%  ;  it  is. .     1.31165103 


hence  the  error  must  be .00034943 

and  the  approximation  holds  good  for  only  3  decimals.    But 
the  correction  can  be  very  closely  computed. 


328  TABLES 

§  376.    Interpolation 

Sometimes,  in  compound  mterest  processes  and  also  in 
mathematical  problems,  we  have  a  series  of  terms,  all 
formed  by  the  same  law,  and  based  upon  another  series.  A 
familiar  illustration  in  mathematics  is  the  formation  of 
squares,  for  example : 

Numbers,  12  3         4  5         6  etc. 

Squares,  1       4  9  16  25       36  etc. 

1st  Differences,           3       5       7         9  11  etc. 

2nd  Differences,              2  2         2  2  etc. 

When  a  series  of  terms  such  as  that  described  above  is 
written  down  in  a  table  opposite  to  certain  equi-distant  num- 
bers called  arguments,  intermediate  terms  corresponding  to 
certain  given  arguments  may  be  inserted  by  a  process  called 
interpolation,  consisting  of  three  steps  : 

(1)  Differencing. 

(2)  Multiplication  of  each  difference  by  a  fraction  de- 

pendent on  the  fractional  distance  at  which  the 
inter-term  is  to  be  located. 

(3)  Application  of  these  corrections  to  the  preceding 

term. 

Differencing  has  already  been  treated  to  some  extent  in 
§§250  and  276.  To  interpolate  in  Table  I,  10  periods, 
a  value  for  2%%,  we  first  set  down  the  two  values  next 
greater  and  next  less,  opposite  their  arguments  (3%  and 
21/2%). 

3%         1.34391638  1,  .      ^ 

^  hdecreasmsf  terms 

21/2%     1.28008454  J     ^  ^    ^    ^   "^ 


or 
21/2%     1.28008454 1 
3%         1.34391638  J 


mcreasmg  terms 


EXPLANATION  OF  TABLES  USED       329 

The  decreasing  series  has  some  advantages  which  make 
it  preferable. 

Continuing  the  cokimn,  use  only  equi-distant  arguments, 
for  4  or  more  lines. 


3% 

1.34391638 

21/2% 

1.28008454 

2% 

1.21899442 

11/2% 

1.16064083 

1% 

1.10462213 

y2% 

1.05114013 

0% 

0.00000000; 

and 

proceed  to  difference. 

Dx       D3 

D. 

z% 

1.34391638 

.06383184  .00274172 

.00010519  etc. 

21/2% 

1.28008454 

.06109012  .00263653 

etc. 

2% 

1.21899442 

.05845359    etc. 

11/2% 

1.16054083 
etc. 

etc. 

Let  this  process  be  carried  out  to  the  6th  difference  and 
we  have  the  following  values,  which  are  all  that  we  need 
to  consider: 

D:  .06383184 

D,  .00274172 

Ds  .00010519 

D;  .00000355 

D,  .00000010 

D.  .00000001 

From  these  differences  any  value  corresponding  to  rates 
between  3%  and  2%%  may  be  determined.  Each  D  will 
be  multiplied  by  a  certain  fraction   (F)   according  to  the 


330 


TABLES 


fractional  distance  from  3%  where  the  interpoland  is  to  be 
located. 

For  the  distance  .5  (which  means  halfway),  the  F*s  are 
always  as  follows : 

R     .5 

F.     .125 

Fs     .0625 

R     .0390625 

F5     .02734375 

Fe     .0068359375 

Multiplying  each  D  by  its  corresponding  F : 

Di  X  Fx  =  .06383184  X  .5  =  .03191592 

D3  X  F.  =  .00274172  X  .125  =  .00034271 

Ds  X  Fa  =  . 00010519  X. 0625  =.00000658 

D4  X  F4  =  .00000355  X  .0390625  =  .00000014 

D5  X  F5  =  .00000010  X  .02734375, 

which  is  too  small  to  affect  the  final 

figure. 
De  X  Fe,     and   following  products   are   also 

negligible. 


Total  correction,  .03226535 

Subtract  from  value  at  3%,       1.34391638 


Interpolated  value  at  23^%,       1.31165103 

By  using  the  above  series  of  F's  (.5,  .125,  .0625, 
.0390625,  etc.),  any  interval  may  be  bisected.  But  the  inter- 
val may  also  be  split  into  5  parts  as  well  as  into  2.  ^  =  .2 ; 
therefore  .2  would  be  Fi  for  the  first  5th,  .4  would  be  Fi  for 
the  second  5th;  and  .6  and  .8  would  be  Fi  for  the  third  and 
fourth  intervals,  respectively. 

We  will  now  give  the  proper  F's  for  interpolating  nine 
values,  each  at  one-tenth  interval. 


EXPLANATION  OF  TABLES  USED 


331 


R 

F, 

F. 

F. 

F. 

.1 

.045 

.0285 

.0206625 

.01611675 

.2 

.08 

,048 

.03360 

.025536 

.3 

.105 

.0595 

.0401625 

.02972025 

A 

.12 

.064 

.04160 

.029952 

.5 

.125 

.0625 

.0390625 

.02734375 

.6 

.12 

.056 

.03360 

.022848 

.7 

.105 

.0455 

.0261625 

.01726725 

.8 

.08 

.032 

.01760 

.011264 

.9 

.045 

.0165 

.0086625 

.00537075 

To  find  the  value  corresponding  to  2.60%  in  the  same 
table:  Since  the  interval  is  .50,  }i  of  the  interval  is  .10,  and 
the  intermediate  arguments  would  be  2.90%  at  .2  distance 
from  .3 ;  2.80%  at  .4 ;  2.70%  at  .6 ;  and  2.60%  at  .8.  There- 
fore, we  must  use  the  F's  of  .8  as  above,  multiplying  by 
them  the  same  differences  previously  obtained. 


.06383184  X. 8 
.00274172  X  .08 
.00010519  X  .032 
.00000355  X  .0176 

The  remaining  terms  are  negligible. 

Total, 

Subtract  from  value  at  3%, 


.05106547 
.00021934 
.00000337 
.00000006 


.05128824 
1.34391638 


Interpolated  value  at  2.60%',    1.29262814 

Had  we  chosen  the  increasing  series  in  our  differencing, 
there  would  have  been  this  variation  in  the  application  of 
the  corrections,  that  the  first,  third,  fifth,  seventh,  and  all 
odd-numbered  corrections  would  have  to  be  added  to  the 
preceding  term  and  the  even-numbered  ones  subtracted. 

We  should  have  differenced  thus : 


332 


TABLES 


D. 

D. 

D. 

21/2 

1.28008454 

.06383184 

.00285054 

.00011260 

3 

1.34391638 

.06668238 

.00296314 

etc. 

3% 

1.41059876 

.06964552 

etc. 

4 

1.48024428 
etc. 

etc. 

The  D's  and  their  products  would  have  figured  thus,  in 
the  first  example : 

2%%    (now  the  basis)     1.28008464 
.06383184  X. 5  +       .03191592 


.00285054  X  .125 
.00011260  X. 0625 


1.31200046 

—  35632 

1.31164414 
+  Y04 

1.31165118 

—  15 


.00000388  X  .0390625  — 

234%,  as  before,  1.31165103 

The  F's  already  given  are  generally  sufficient  for  any 
practical  purpose,  but  even  if  a  very  unusual  fractional  rate 
requires  computation,  the  F's  may  always  be  worked  out 
by  the  following  formula : 

Fi  is  always  the  distance  from  the  first  value,  expressed 

decimally. 
Subtract  Fi  from  1,  multiply  Fi  by  the  remainder  and 

divide  the  product  by  2 ;  this  gives  F2. 
Subtract  Fi  from  2,  multiply  F2  by  the  remainder  and 

divide  the  product  by  3,  giving  F3.    And  so  on. 


EXPLANATION  OF  TABLES  USED       333 

Observe  that  it  is  always  the  original  Fi  which  is  sub- 
tracted from  1,  2,  3,  etc.,  and  that  the  divisor  is  always  the 
number  of  the  F  sought. 

This  will  be  plainer  in  symbols. 


F.  =  F.X(1  — FO 
F,  =  F.X(2  — FO 
F.  =  F3X(3  — FO 
F.  =  F.X(4  — R) 
etc. 


Fn  =  Fa_x  X  (n  — 1  —  FO  -  n 

The  F's  already  given  should  be  worked  out  for  practice 
by  these  formulas. 

As  an  example,  we  give  the  F's  of  .24. 

Fx  =  .24 

Fa  =  .24  X  0.76-^  2  =  .0912 

Fa  =  .0912  X  1.76 -f- 3  =  .053504 

F.  =  .053504      X  2.76  -^  4  =  .03691776 

F5  =  .03691776  X  3.76  ^  5  =  .02776215552 

Where  the  rates  given  in  the  tables  are  more  than  1/2% 
apart,  interpolation  is  not  practically  useful. 

§  377.     Table  VI — Reciprocals  and  Square  Roots 

This  table  gives  the  reciprocals  and  the  square  roots  of 
120  of  the  most  necessary  ratios  of  increase. 

The  ratios  begin  at  %  of  1%,  and  increase  by  40ths  of 
1%  to  3%  ;  by  4ths  of  1%  to  7% ;  and  by  1%  to  10%. 

The  second  column,  composed  of  reciprocals,  gives  the 
present  worth  of  $1  payable  one  period  from  now,  like  the 
second  line  of  Table  II.  It  is  used  for  the  purpose  of  dis- 
counting by  multiplication  rather  than  by  division,  the 
former  operation  being  much  easier.    Any  reciprocal  may 


334  TABLES 

j 
be  tested  by  multiplying  it  by  the  ratio  standing  opposite, 
which  will  give  as  the  result,  unity. 

The  third  column,  composed  of  square  roots,  gives  the 
equivalent  effective  ratio  for  a  half-period.  Thus  for  an 
obligation  at  6%  semi-annually  the  ratio  of  increase  is  1.03. 
If  a  quarter  of  a  year  (a  half-period)  has  elapsed,  the 
amount,  if  scientifically  treated,  is  not  1.015  as  used  in 
actual  business,  but  1.01488916.  If  the  loaner  were  to  re- 
ceive 1.015  as  the  amount  after  three  months  and  reinvest 
at  the  same  rate,  he  would  have,  at  the  end  of  the  half- 
yearly  period,  not  1.03  to  which  he  is  entitled,  but  1.030225 
(=1.015^).  But  if  he  receives  1.01488916  and  reinvests 
for  the  other  quarter  at  the  same  rate,  he  will  have  at  the 
end  of  the  half-year  1.01488916'  =  1.03. 

In  other  words,  if  .03  is  the  rate  for  each  period,  the 
equivalent  effective  rate  for  a  half-period  is  .01488916.  To 
receive  or  pay  3%  each  half-year  is  exactly  the  same  in 
effect  as  receiving  or  paying  1.488916%  each  quarter. 

Intermediate  values  in  the  second  and  third  columns  may 
be  readily  found  by  interpolation,  usually  requiring  only 
one  F. 


CHAPTER  XXXII 

TABLES   OF  COMPOUND   INTEREST,   PRESENT 

WORTH,  ANNUITIES,   SINKING  FUNDS, 

AND  OTHER  COMPUTATIONS 


335 


336 


TABLES 


^ 
^ 


COON  T-HCMiOOr^  "^voo^rj-r^ 

VOOO  CM-^t^ONON  vovoooooo 

u-)QC^  00  fO  u-i  CN)  CVJ  rfvOOOT-HfO 

C^O\'-i  00\000^  OOCOOO'-Ht^ 

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OOOr-i  T-H  ,— I  ,-H  CO  <M  CMfOcOcO''^ 


OOiOOOOOO 

00 Tj-^  OS 00 

Tj-00CMlOO\ 


■^loooom 

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•-H  00  r^ '-I  CO 

00  ON '-'■^00 

fOt>»C0  VOO 
^  vOt^t^^OO 


OOCMCOO 

■^roo>otv 

Tfo\oo\c 

ONOOOOTfTj 
iOO'^ONTj 

OOOnOnOnC 


rOr-HfO  ^TfO'-lOO  TflOONr-lr^ 

^  VOOCO  t^tOONT-HTj-  COOO\VOCO 

>!.  O«r)00  t^cgroro^  O'— '-^cooo 

NT  toiOOO  ^OOiOOOt-^  CMCOQTj-Tl- 

CMTtVOQN  '-htJ-^OnCM  '^rft^OcovO 

OOOO  T-H  r-(  r-H  ,-H  CM  C\J  CM  CO  CO  c^ 


O^-^MDt^  OvOCMt>^-0  CMCMOOOni/ 

00 -^  ON  1— I  CO  CMVO'— 'COU-)  covooOOnC 

VO'-HCMt^O  0\OCMt-HV0  vOOMOrfts 

OrO-^OOr^  OCMCMCOVO  'rtOO^-rtO 

CM^f^iOr-i  io\0"^CMr^  '-I  CO  IT)  lo  T) 

V0I>*0nCMV0  QtO<— lOOm  rfcocOTj-vr 

OnCMioOnCM  ^OncovOO  "^OOCMvOC 

co-<^-*tJ-io  lOiOVO^t^  t^t^ooooo 


VO  OCMt^OOr^  CM'-"a\C0\O  ^J- --h  cm  lO  in.  O^^tN-VOiO  O\r-i00»-<0 
t-H  OO'^VOcotO  "^cot^vot^  cot^'^CM'-"  '^covOCMCM  On  i-"  tJ- CM  VC 
OOCM  0(M>r>ON(N  TfTt-^vOOO  OOO^vOt-i  t^vOONONt^-.  ioOOvOtJ-' 
^ VOOO'^'^^  ^ —     ~   '-  ^        


CM 


■^  VO  r^  On  CO 


oco  oovoooiooN  ONt^Ttor^  NOi 

SCNjTd-  O'-HVONOO  ONCorovOTt  001 

T-iCM  '<*•  VO  00  T-i  uo  OOCOOOCOCJN     >J^ '       _  _ _        _ 

CM'^VOOO  OCM'^t^C^N  "-HTt-voON^  ^t^OCMiO     00  i-h  rM^  O     rftN.O'^t^ 

OOOO  ,-1 T-H  ,-1  T-i  ^  CMCMCMCMco  co  CO -^  ■<*■  tJ-     '^mmxno     NOvOt^t^fN 


vOrfOOOa 
OcovO'-Hir 


fin 

I 


o 

H 

o 


iO'thco     VO  lO  u->  00 '-•     ONt^i-HOCM     voirj^^o     O  t-i  ^  VO  O^     -^OvOOCV 


^ 
^ 


^ 
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lOCO'-Hf^CO     Tj-f^cotoOO     OOcO'-i^ 
VOCMC^I'-Ht^     TfOvONONVO     r^OvOO 


OCMiO     '-lOCMOOOO  r^LOCO00'-^  CM  CM  CM  co  tJ- 

lOCO'^OO     VOlN»^00ON  Tl-CM-^^ONON  CMOnO>JT^ 

t^Loco^     OOnOvOOOO  On  O '-h  cm -"^  t^  ON  co  vo  O  -^  On  rf  o  vo    CMOnKi/) 

^              -"^  VO  OOr- (coior^  On  T-H  Tt  VO  On  >— i  co  vo  On  t-h     -"^voOnCV) 


.     CMOOt^Tj-r^  iovocoonvc 

to     OOvooOt-iCM  QCMt^CMt^ 

t^coCMVOrJ-  OOOOtOr-Hii- 

t^iot>*coTj-  On  0\  Tt  Tf  « 


r-HCOLOt^ONOCM  _  __  

OOOO     o '-"'-' --H '-"     '-iCOCMCMCM     CM  co  co  co  co     "^  "^ -^  Tf- lo    mmiovOvC 


00  lO 

CO>r) 

IT)  00  CO 
CMt^vo 


•IT)       rlCOOlOi-H      irjCOOOOO' 
It^        O00t^»-H00       CO  irj  ,— I  r-H  U- 

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CM  a 


cOtJ-CMON  OOOOOCMvo  CMioOOO 

_         ,^tJ-OnOO  Ttrti-^miO  cOoOCM-rt-iO  lOiOVOr^O  Tt-O'-.-. 

CM  VO  CO     CM  Tt- OO '^j- CO  lOONVOmtN.  CM  On  O  co  On  OOOiOCOvo  CTNt^OOCOO 

lOOLOi-t    r^coONVOco  Qt^tOCOr-H  oo< ~     —     " ■"     ~ 

r-(CO^vo     f^ONOCM^  vSwONr-HCO  ipv^      -_..        . -.          

OOOO     O  O '-<'—' '-<  1— 1 1-H  t-H  CM  CM  CMCMCMcoco  CO  co  CO  tJ- t^-  rf  tJ- t}- in  ir 


iOOtJ-  lOOOt^OOO  COCMCMtOiO  OOiOt^ONi-t  coOOOQnij^  rJ-mCMO'- 

CM^>.co  r-H  ^  Tt  r-i  r-i  oOCMiOONt^  t-Hvoi-HCOVO  CM  t^  "^  o  o  oncooncocv 

VOOiO  CMCOOVOCM  O-^-^co-^  ONON00r>.ON  r^CMOQf^'-H  CMiOOCMCv 

lOt^Tf  OOOOtOOOON  t^CM^OVOtO  CMOOCOr^O  COVQOO'-Hm  ON'^'-'ONO 

lOt-HONQN  OCOOO-'^CM  CM-^t^CMON  OOOO'-'iOCM  ~               "  '"        -- 

CMiOt^O  ■"^t^O'^OO  CMVOOloOn  -^CTnloovo 

T-iCMcoiO  vot-^ONOr-^  co-^vot^OO  O^coiovo  00  On  r-*  co  rj-  vd  00  On '-' cr 

OOOO  000'-''-<  T-H  ,-1  ^  ^  T-^  CMCMCMCMCM  CM  CM  co  co  co  CO  CO  co  "^  rt 


OOCMt^co     i-hCMioOnvC 

CM  oo  Tf  o  r^  


COtOCf 


rH     IT)  VO  lO  T-H  ^N,     COiOCOOO'-i     VOt^COOOiO 
O     0»-"Cor^CM     »-HCOOCMC^l     OnvO'^'^ON 


•Tl-voCMm     O' 


'-i'^  OO^ovOiO  CvJ  00  to  CO '^  OOOO'^t^OO  _ 

OO  1— iCMcotooO  CMVOCMONf^  vo  t^  O -^  O  On  On '-h  VO  CO  coioOOnC 

1— I  CO  VO  Oto,— lOOVO  vovOOOOtI"  OniocO'— i»-h  i— i  co  1>» '— i  t^  rj- CM  CM  CM  iT 

OOO  '-Hi-HCMCMco  -^tnvooOON  OCM"^VOOO  O  CM '^  t^  ON  CM  to  00 --' -^ 

'-HCMCO'^  tovOt^^OOCTN  O'-'CMCO'?*-  vo  t^  00  On  O  CM  co ''t  to  VO  00  ON  O  CM  cc 

OOOO  OOOOO  T-H  r-H  T-H  T-(  r-(  ^  r-H  r-^  ^  CM  CMCMCMCMCM  CMCMCOCOc: 


00 

CO 
000 


Ot-^CMco-*    lOVOr^OOON    OrHCMfO-*    iri\ot-^COO\     O-^CMcorf    »J^'Ot>oqON 
^^T-trHf-t     ^T-trHr-H,-l     CMCMCMCMCM     CM  CvJ  CM  CM  CM 


f 


COMPOUND    INTEREST;    OTHER     COMPUTATIONS 


337 


vot^ONOO^ 

mor^  00  fo 

t^  O  f*^  OO^o 

O'—CvJCVJCO 

cvj  c4  csi  csi  cvi 


a\<Mooqoo 
lo  m  cx)  Cvj  Tj- 

O  rOr^COt>N 

c4  cvj  cvi  cvi  cvi 


Tt  roOCX)00 
OCI-CNOO 
fOOiOOOO 
vOCAOnCNO 
O  '-'  OMJ^OO 
LOCVl  O  '— '  <^ 
CO"^C\J  OnVO 
vor^oooo  On 
<\i  (\i  c4  cvi  og 


OOVO  fOVOO 

rq  CX3 '-<  lo  CO 

fOOt^  ON  vO 
OiOONOOf^ 
ON  00  vO"^  CM 
t>x  CO  r-.  T-H  ro 
CO  '— '  ONt^iO 

p   t-H    T-H  CM   CO 

CO  CO  fO  CO  fO 


CMPOiOVOVO 
t^Ot^CMOO 

00  CO  0\  00  CM 

or^ooioo 

i-.t^t^ON'-i 

r^oooNt-^CM 

CO  00  On  t^  CO 

TfoqcoONvq 

COfO'^TfiO 


coCM-^coio  m 

TtOOCM  COOO  CO 

t^t^  rfvOco  VO 

O  ^  vOiO  O  '— I 

cmioonoovo  r>. 

CM  Q\NO00^  CO 

X^OIOCMtJ-  .-1 

COCM^CM-^  00 

vdKodoNC)  '-< 


Tj- <7\  CO  Tt- lO  vOCMOnOnO  I^ogoOO'^  0\0'-"-'0  OOOnOOO  lO  O  CM  CM  ^  O 

Tj-r^OCOTf  lOrt-CMCJNON  ONt^CgTtTj-  coonconOio  rl-rl-r^OOTt  OcOi-hOOCM  CO 

cOTfcoOON  covOCMiOCM  COOmO"^  vO  lO  co  ON  ^  NO  r^  rt  vO  ,-i  TfiO^rN.0N  no 

CMOONOTf  ON'-htI-OnO  OOOOOOnNO  m  On '— '  co  O  -rf  CM  co  CM  t1-  t^  Tt"  On  lO  O  "^ 

cocM-^ONOo  t^oooN'-'No  «-ionOcmoo  x^onvono^  oO'-'CO'-h  r^'-Ho^c^^>.  cd 

OncoOOcoO  OOl^t^ON^  iOC^nocot-i  »-HCMtOONiO  CMOOt^r^  vnOt^r^ON  ■^ 

tKTncooOco  I^CMr^CMOO  coCOrfQNO  CMOO-^Ot^  TfOO"^"^  OCOCNJCDJ^  »0 

OnOnOO'-'  t-iCMCMCDco  tJ- rf  lo  VO  VO  r>»  t^  00  ON  CTn  O  Tf  00  CM  t^  CO  On  NO  rt;  CM  CM 

.-J»-^CMCMCM  CMCMCMCNJCM  CM'cmCMCMCM  CMCMCMCMCO  cococOTfTt  \n\n\6t^o6  On 


OOCMOnOco 
lOOOuT^O 
t-iCOO'-ivo 
NOCO^l-cot^ 
CO  U-)  U-)  CM  NO 
i-Hl-^  rj-  CM  O 
i-H  Tt  CO  C^^  NO 
OOOOOOOvON 


lOTj-ONCTst^ 
lOCOOt^t^ 

ONt^iO  COtJ- 
OOOOOOOn-'^ 
OOOOVOCMt^ 
OnOnQCMtJ- 
On  CO  55  CM  NO 
OnO  O  '— '  »-< 
•-h'  CM  CM  CM  CM 


OCMCM  '-I  O 
00  CVI  t^  CO  O 
O  lo  On  Tj-  On 
CM  CM  CM  CO  CO 
^4  CM  CM  CM  CM 


1— (  On  '— I  On  On 
CMCMiOfOl^ 

Tfr-H  COO^ 

00  VOCOOOO 

t^  NO  NO  r^  00 

CO  00  CO  00  CO 
Tt  r^  LO  li")  NO 
CM  CM  Cvi  CM  CM 


coi^ONf-iCM 

ONorN».-HCM 

OOOOCOOO 
00  coco  CM  lO 
LOI^  Ol-OtO 

^^,-hCMON 
On  r^  00  CM  c^ 
NO  On  CM  NO  ON 


OOrtOO'-'NO  NO 

u-)ioCM  C0 1— 1  T:^ 

i-it^OOTfNO  Tt 

tj-oococtmo  cm 


CMCMCOCOCO    ■^Tj-miONO    t^ 


CO  CO  On  CO  CM 

"-H  T-H  Tj- CM  lO 

OCTncoCM  ■'^ 
OTt-HOCM 
OOCM  (Mt^t^ 
CM  CM  CM  CM  CO 
OOT^^t^O 

Not^r^t^oo 


Ot^ON^Tf 

r^  CM  00  "^  00 

Cht^NOOO  T^ 
COOOOcot^ 
CMrfOcOi-i 
LO  t^  Q  CO  l^ 

CONOOCONO 

OOOOCAONCTs 


Tj-O-'^OOv 
cocoCM  On  T-H 

r^ioNOOO 

ON  CM  NO  coco 
lONOCMLOTt 

'-"  NO  CM  00  lO 
O  CO  t^  O  "^ 

CM  CM  CM  CM  CM 


CMOOOnCMO 
CM  CM  00  t^t^ 

VOt^t^OOr-H 

t^l^Tt  OntT 

ON^-i  o^ooo 
CM'-hOOnON 
00  CM  NO  On  CO 
'-H  CM  CM  CM  CO 


coiooo-^t^ 
On  00  CM  t^  CM 
OOt^NOiOOO 
OOC^Ii-tCMOO 

r^iooO'si-CM 

ONO'-^OOOO 
CO  On  CO  00  NO 
COIOOOOCO 


OOCMOO'^  Tt 

ONONTj-OOCM  C?N 

O  '-Ht^OCO  ii-> 

'-H  ONCOOOO  u-> 

inco-rr  coCM  T-i 

cONOCMOt^  00 

t^QNONOON  NO 

NOOCOt^'-H  NO 


CMCMCMCMCM     CM  CM  CM  CO  co     CO  "*  tJ- rj- lO     lo 


CMCMCMOOt^  CMTj-COOOiO  »-«  00  CM  CM  CM 

CM"^co.-ico  covoNOCMCM  "^  vo  t-h  00  O 

ONOtJ-OnNO  ^C^nOOOO  COCOr^ONCO 

OOCMCMt^ON  OOcot^ON'— I  T-HCMrM^co 

0"^fOTl-0\  OO^t^r^CM  CDCMOOOOco 

coNOOtI-CO  coO\"^Qr^  ^r-iCONOiO 

NOOO'— 'COiO  OOOCOVOOO  ^TtNOCTvCM 

lOiOVONONO  vot^t^r^t^  COOOOOOOOn 


^^OOnno 
OCM'-hCM"^ 

CONOCAOOO 

r-HCMt^r^CO 
CM  to  CM  Tj-  r-H 

•^COCOCOTt 

to  00 '-<  rj- t>x 
C?nOnOOO 
>-;  T-I  CM  CM  CM 


CMcoOni-hno 

Tj- Tt-  i-H  Tl- lO 


CMCMCMCMCM 


'-iOnOO'-h'^     lO 
tV.t^COlOr-1      VO 

t-HCMOOOOCM     VO 
On  NO  t^-^  On 
ionoOnOno 
-^O-^OOrf 

■     t-H   ,-H        CO 


to  (_ 

O  CM  to  00  i-H 

CO  CO  CO  CO -^ 


R 


NOcO'-HTfON  t^CMCMOOtN*  VO  »-*  t>»  to  i-H 

cotoiO^CM  OOOO^t^OO  Ttt^l^OOCM 

COOOOCMNO  tococONOr^  C?N-^NO00rf 

T-HiococOVO  COtJ-OnOOCM  ^vONOCMto 

NOr^'— it^to  voOn-^CMco  no '— '  CTn 

.— lOvooNOto  Tj-cocococo  co'^l-T^ 

toNOCOOCM  TfNOOOOCM  Ttvo     " 


TfTfrflOlO      tOtOLONONO      NO  NO 


CO 

OOOCM 
\Ol^t>^ 


Tf^vVOtOON 

»-iONOoot^ 

NOt^  CO  Tj- VO 

"^O-^to-rf 

OnOO  OCOQ 
OOOCMtoOO 
Tl-t^  ON  T-H  CO 

t^t^t^oooo 


t^  Oiot^l^ 

cot^coOO 

CMCO00r-H5N 

OCM  '-H  CMOO 
y->  Ot^CMtO 
NOCOOrJ-00 
OOONt-hCM  CO 
'-h'  »-;  cvj  CM  CM 


tOONONVOOO     CM 
corfr-Hr- 
OnOOO' 
r^TfNooO' 

CO  Ot>>toio  VO 
tot>s  OOOCM  Tt- 
CMCMCMCOCO     CO 


CMOOOOnOn  vO00t>^Tti-t  coi-xOvOvr^  to  to  rj- 00  Tj-  CMt^OONt^ 

ONTfNDOON  t^f^-^CNlto  t^coOOt^iO  t^OOrJ-Oco  00  to  t^  Tt  CO 

OOt^OONO  CMOONOt^CM  CO  CM  On  t^  r^  O  00  co  NO  00  '-i  Tt  NO  NO  CO 

TfCM-^ONt^  ONOt^CMCM  NOiOCOt^'— '  r-(  to  VO  CM  rj-  CO  CM  ON  NO  NO 

OOcoONVOto  NOt^OtO^  OOX^t^ONCO  OC^CMCMco  vo  to  NO  co  l\ 

tv^^Tl-OOCM  NOOtOON-"^  OOcoOOcoOv  tT  O  NO  CM  00  Tf  00  NO  On  NO 

Ti-NOt^coo  '-icoTfior^  ooOr-icoTi-  noooont-hcm  -^cM'-'OO 

CO  CO  CO  CO  tJ"  Tf  TJ- Tt  tJ- T^  Tj-tOlOtOtO  tOtOLONONO  VOt^OOONO 

^^^^r^  ^r-ir-i^^  ^^'^^^  ^^^^^  ^^^^CM 


tN.CMt^ 

Tt-CN]  C?\ 

00tOC3N 
CM  '-'OO 

CDnvoOn 
O'-'CM 
i—CMco 


t^lO  CO 

NOtO  00 

CMt>x  CO 

CO  CO  »-H 

NOlo  00 

OOco  rf 

Tj-t^  O 

Tl- to  t^ 


CMCMCMCMCM     CM 


O'-iCMCOTt     tONOt^OOCN     O^-hCMco 
cococococo    cococococo    '^'^rf"^ 


Th    tovotv.00ON    OtoOtoO    tOQlOOtO 
-^     '^'^-^"^"^     totONONOt^     t^OOOOONON 


338 


TABLES 


VO 

00  »-"  vot^N  VO 

Ovot^vovo 

On  00  C3NtoO 

t^OCM  VO-^ 

CM  VO  -TfOO, 

On 

tOi-iCMOOv 

t^to-^CM  Ov 

r-H  vot^  '-'to 

Tf  VOTl-VOVO 

t^  ON  CM^  On 

VOVO 

tOOvOCO  00 

r^oo  VO  00  CO 

CO -H  CM  ON  On 

to  cot^C^'* 

O  CM  tn  \0  t-^ 

T-Hl^ 

tJ-OnOnCMO 

to  to  t^  CO  ON 

CO  vocOTt  ro 

r^ CO  ^00 00 

•..o 

vOO  "^ 

oocM  T-^  a\CJ\ 

to  CO  t^  CO  to 

T— 1  lo  to  r^  ON 

OOcoco  VO  CO 

^ 

roi=H(N 

OOOOcoroOv 

poo  CM  CM  O 

a\Ov^-<<o  VO 

VO  O  C^J  -^  to 

t-.  On  CO  On  00 

1— 1  On  cvj  r-H  00 

VO 

VOCN)  On  VO 

CO  T-H  OC3N00 

On ''^  On  to  CM 

OOnO— 'Tf 

0\  -^  CV)  Y-n-i 

T-H  ^  T-H  T— 4  r-5 

fOTftoto\q 

t^OOO  '-'CM 
i-;r-;CMCMCM 

COtovOCO  o 
CM  CM  CM  CM  CO 

CM  CO  vqoq  p 

CO  CO  CO  CO  tJ" 

CM  to  CO  '-'  ■>* 
•Tt  ■^■^tou-j 

m 

VO-^CM  -^CM 

CO  vOcO'^O 

00  On  CM  coo 

'-' CA  CM  VO  Tt- 

^%^^% 

CM 

to  VO  -^  rj-  CM 

OCOCO  ^  VO 

'-'to  CO  CM  CO 

r^totN.t^  On 

lOVO 

^toOtoOO 

Tl- C?N  VO  CTn '-' 

COtI-COOnO 

t^  CM  O  CO  ON 

"^CM  VO  CMO 

CMO 

00  ONOtoCM 

Ovcoiorf  CO 

00  cooovo  a^ 

c^j  r^  ^  '-I  to 

On  NOVO  CM  ON 

to  t^  to  CM  CO 

^ 

to  VOlO 

OjO-HTtrO 

ONOO  O  VO  o 

CV)  On  CM  to  O 

COVOTtr-.^ 

CMt^to 

voot^r^'-i 

CO  O  to  to  On 

00<MCMVOVO 

CO  to  to  »— 1  lO 

vOto  crj  o  VO 

to 

to  O  t.^ »— 1 

r^Tj-  or^to 

CM— 'On  00  r^ 

t^OOOvOCM 

to  CO  CM  t^  CM 

OOto  COCM  v-H 

.  p  '"1  ^  ^. 

CM  CO  ^  -^  to 

vqt^f>.oq  ch 

p  >-;  CM  ''^  to 
CM  CM  CM  CM  CM 

vq  t^_  On  O  CM 
CM  CM  CM  CO  CO 

CO  to  l^  On  r-i 

CO  CO  CO  CO  Tf 

— 

:2S 

rj- CM  ro -H  Tj- 

CMtocoOCM 

TttO  '-<t^  r-H 

(N  vo^toco 

vO-HVOONOv 

H 

^Q 

Ov— loovo-^ 

Tj-OTfr-iav 

^^cor>.co 

Or-HOCOOO 

Tl-  O  >-0  0\  Tj- 

e^ 

lovooo 

'-H  O  '-I  Oto 

OVCO  ^  vO'sf 

CMOVOCOO 

Tf  T-H  cvi  VO  CO 

Tl-  On  CN  CJN  VO 

C/3 

:i^ 

CMVO-^ 

OOVOVOOOV 

voto  CO  On -^ 

00 1^  i-^  r^  VO 

-HTt-tovo— ' 

CO  t^  O  On  c-j 

M 

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COMPOUND    INTEREST;    OTHER     COMPUTATIONS 


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CM  CM  CM  CM  CM 


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a: 


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340 


TABLES 


vo  00  u->0\ 
vo  t— I  OOiO 

On  0^  On  C5\ 
"OCJOO 


0\t^"^txvO 
eg  00  CVl  u->  CO 

rl-^Ovo"O00 
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OO  CM  CMt^t^ 

COCN  ^OO 
OOMDrfCgO 
00  00  00  00  00 


QOOOiOOO 

00  "^lo  or^ 

ONTfioCMCM 

^  CM  CO  to  r>H 

OOvOTf  CNO 


ooooo   ooooo 


"^  0\0  0\l~^ 
u-5  rf  to  lo  r^ 

VOCNJ  0\^  CNJ 
Tf  vo  ^  r-c  lO 

ocor^x  ^  to 

On  t^  to  Tf  CM 

vo  VOVO  vo  "O 

oc5oc50 


CVJ  rOOO  vo  CO 
Oto  O  VOCM 
■—I  On  00  vo  to 
vo  to  to  to  to 

ooooo 


OnCvI  CO  00  to 
to  !>.  r^  t>^  cvj 

O  "^  0\t^  r-l 
ONCOONt^VO 

CO  eg  CO  00  vo 
ONVOCOOOO 
cocg  ^  OOO 
to  to  to  to  Tf 

oooc5o 


< 


CM 


^ 
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T-HOOOO 


to  CVJ  t^  VOtO  »-H 

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CO  CM  T}-  OnOO  '-I 

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00  t^O  t^  ONto 

rj-  rj-ioto  vo  00 

y-l  ONt^tOCO  '-H 

ON  0000000000 

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c  )  On  00  to  In. 

r-l  ONTfOCO 

O'^I^On'-h 
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to 00  vo  00  CO 
8  CM  to  00  CM 
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oo  t>s  r>.  tN.  t^ 


OOOCMCO-* 
(NOO  '-•VOOO 
vOtoCMl^rt- 
CM  vOtot^  CO 
CMtJ-OOnCM 


VO  O"^  C7\to 
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t^t^  vo  vo  vo 


r^oot^Tj-co 

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votOTj-i^  vo 
r-H  ^Cgco'-^- 
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OVOCMONVO 

TfCNJ   r-lONOO 

vo  vo  VO  to  to 


OOOOO  OOOOO  OOOOO 


CTM^t^OOco 
C0  0\  '-H  00'-< 
vo  ON'-'  CO  CM 
"^(N  ON  CM  CM 

cor^  CO  CO  to 

cOOCOVOTf 
t^  vOt}-  COCM 
to  lO  to  to  to 

o  o  d  o  d 


vo  00  CO  CO 
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co^  cooo 

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t-hOO 
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t^ON 
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vo  On  to 
tori-r^ 

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t^tO  CO 

CO  00  00 


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00  00 


00  CO  CM 

t-H  too 

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odd 


CO'-H  voi^vO 
t^  00  to  CO  r^ 

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cooo  -^  O  vo 

Ttev^  t-H  ooo 
t^t^t^t^  vo 

d'd>c:>d>d> 


CO  CM-*  CM  ON 

coOOOChTj- 

T-H  to   0\  to   1— t 

t^vt^  coto  CM 

ONt^oo  "-ir^ 

CM   ON  vo  Tf  r-H 

lN.tOTj-C0CM 

vo  vo  vo  vo  vo 


tN.00TtlO.-< 

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COt^  VOt^»-< 
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vo  to  to  to  to 


oorN.00  ri 
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8  t^  to  00 
On  CO  to 

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OOVOtJ-co 
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rf-rfOOt^tO 
lOtO  t^tO  CO 

CM  CM  CO  t-H  T-H 

On 

vo   r-(  to  O  to 

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ON  ON  OO  00  00 


t-tOOOO     OOOOO 


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t^CM 
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CO  00 
do 


ooooo 

00  CM  ON 

r^  '-t  Ti- 
to ON  vo 

ooco 

CM  COTf 
»-H  ONCO 

oot^t^ 


ONi-llOO'-t 
tOCOOONO 
Tf  VOVO  ON  Tf 

t^'-HOOt^ON 

oovotor^  ^ 
orN.Tf  »-H  On 

t^tO  Tf  CO  •— I 


OOOvOOt^Q 

toco  eg  r-io 

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00  vOt^<3^Tt• 
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t^  vo  vo  vovo 


OOO    ooooo    OOOOO 


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00  vo  >o  to  Tj- 

Tf  COCM  '-lO 

vo  vo  vo  vo  vo 


o 

w 

H 

Pi 
o 

H 

w 

CO 

w 


i>NtooNco  coononcm'*  cocoegcMoo  o^j-vocm^ 

vOt^ONCM  cOr-Ht^^CM  CMCM^tOeg  toOCMto-^ 

'-•'-'VOrJ-  OCMvOt-iCvj  t^cot^t^Ov  ^^to,— it^ 

V..O          CMvo.-<oo  voTfcv)r-HON  vococoeg-Ji-  tocoooi-io 

e^             CMVOCO'-H  CVl>OOtN.to  vOOnCOOOO  COOCOCTnVO 

Vs              tOOVOCM  OOTt-1-it^^  ^oOVO'^r-i  ONOOVOrtCO 

►X            COt^tOTj-  CM^-HOOOt^  vOtJ-coCM'-h  CTNOOl^vOto 

^CTnOnOnON  CJnOnOnCOOO  00  OO  00  CO  00  t^t^tv.t>.t^ 

idic^dd  dcSdcici  cDcid>ci<d>  dddcScS   ddddcD 


C\1  to  CO  00  CM 

r^C3NVOOC7\ 

ot^t^t^<^ 

I^CTnOO  co-^ 
Tf^vO  Oto 


COCM  Tj-iOl^ 

CO  to  r-^  CM  00 

to  O  to  On  00 
OCM  OOONtO 
CMO  OnO^O 
ON  On  00  On  q\ 
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vo  vo  vo  vo  vo 


CM  vo  CO  00 
CO  oco  CM 
""t —  oo^ 

voTJ-T^to 

t^tO  CO   r-H 

00  f^  vo  to 
On  Ot  On  CTn 


VOOO  CO  to  On 
O  CO  CDNTf  vo 
r  >.  Tt-  to  00  o 
l^  I^  »-i  On  CM 

r>. ,— (  r^  CO  CM 

onoo  votoTt- 
co  eg  '-I  o  C7N 

On  On  On  On  00 


covoo 
ONTl-vo 

ot>>.oo 
coi^o 

1— I  CM  to 

COCM  -H 

oot^vo 

00  00  CO 


^OOOO     ooooo     OOO 


CJnON 

VOO 
CM  00 
t^  vo 
00  CO 
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to  Tt- 

00  00 
cSd> 


00  to 

t-ico 
covo 

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CMTtvo 

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voooo 

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vo  vot^ 
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to  »— I  vo  cor^ 

to  00  On  too 

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ooNoegrf 

cocotovOtN. 

coeg  r-ioqj 


ooooo   ooooo   ooooo 


^ 


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On  Tt  to  eg  CM 

vocMoe^ioo 
to  to  00  CO  On 
V0r^r-.00co 

■^  Ot^Tf  CO 

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tOTf  CO  CM  <— I 
<?N  On  On  CA  On 

c?  <z>  <z>  d  d> 


tOCM  COOt^ 

ONt^CM  vo  On 

VOCO  On  CM  eg 

OOCMr^vo  vo 
CM  COTI- vo  C?N 

tOVOt>*  00  C?N 

o  c^oot^  vo 
cjnoooooooo 


t^voCTN'-^CM 

rj-eg  Ttcoc^N 
On  '— ir^r^o* 

Tj-CMr^  r-H  CO 

cococoofN. 
'-^eg  rfvor^ 
votoTj-  roeg 

00  00  CO  00  00 


t^r^i— I  cdnco 
Tt- ,-( CM  r^  T-H 

T)-OVO  ^  vo 
Tl-  CO  CTn -^  vo 
to  ""^  CO  "^  to 

On  '— <  CO  to  t~>. 

'-I  -hOOnOO 

oocooot^t^ 


Tj-voCMt^tO 

Tf  Ol  C?N  to  r-l 

00  t^  CO  to  CM 


ooooo   ooooo   ooooo 


O'-hCMcO'^    tovot^OOON    O^CMCOTl-    iovor>.00ON    Oi-iCgcOTh    iovOt^OOO\ 
,-<  ,-H  ,-H  .-H  ,-t    T-(  T-H  ^  rH  r-(     CMCMCMCMCM    CMCMCMCMC^ 


COMPOUND    INTEREST;    OTHER     COMPUTATIONS 


341 


C^l  Tl- O  CNJ  u-> 

■>!4-  T-HI^  O  O 

vo  10  CO  eg  .—I 

t^  vo  >r> -rj-  CO 
'^  Tt  Tf  Tt  T:t- 

OCJOOO 


t^  OWO  00  -^ 
fOOO<Nt^ 


vo  a\oooot>s 

O  vO  -^GO  CO 
CO  -tT  CO  CO  o 
Tf  COTJ-OO-^ 
CNJ  CO  Tj- lO  l^ 
t^  ^O  l-O  Tt  CO 

CO  CO  CO  CO  CO 


ON'-' 
CMfVJ 
coco 


TfVONO 

O.'-Ht^ 
Cq  T-HIO 

covOCNJ 
COLO  00 

^oo\ 

CO  CO  CN) 


r-HoqoM^oo 
csj  10  U-)  u-j  m 

CM  o  com  CO 
T^m  coooio 
0\  '-'  CMOOm 

OMn  CM  o  i^ 

CM  CM  CM  CM  —' 


00000  00000  00000  00000 


0\r^ coonco  t^ 

Tf  »o  MD  t^  r^  CO 

T-i  Tf  rf  irj  O  t^ 

CO  o  OMor^x  Tj- 

OM^  Lo  CO  r>*  vo 

VOOO  CM  <X)u-)  T^ 

u-3  COCM  O  ON  CO 

'-"-i^^O  o 

odoCJCD  CD 


OO^COOO'-H 
O  O  C0l0'!i- 
0  CM  CMiO  ^ 
C/J  OmJ-j  10  Ch 
ON  'O  OCX)  CM 
CM  ^OCTnOn 
'-'  O  ON  1^  o 

U^LO';^r;^  Tt 

CDOCDCDcrJ 


OCM 
VOO 

ONO 
CJnGO 

coco 

OCD 


00OC50 
Ot^CM 
CM  CM  10 

ON  CO  CO 
On  coco 
CO  ChON 
CO  CM  ^ 

000 


VOTfNOlOCO 

t^  m  >— I  CM  >o 

10  On  CM  ON  NO 
•<^  o  r^  CM  1^ 
vo  NO  t^  ^  ^o 
O  '-'CMTj-io 
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tT  Tfcoco  CO 

OCJCJOO 


•— •  O  On 

0000 

C?NLO00 
OCM  — ' 

Tf  CO"^ 
NO  10  10 

COCO  CO 

<od>d> 


00  CM 

10  CM 

C»CM 

CO  NO 

'^fO 
coco 


00  00  NO  NO    -,N 

O  CM  10  CVI  O 
NO  iri  00  CM  CO 
CM  r-(  Tt-  Tt  10 

t^   T-"-H  Tf  NO 

CO  Tf  CO  to  O 
cvi  On  NO  CO  '-' 
CO  CM  CM  CM  CM 


1-1  coOOt^  On  tJ- 

On  On  CM  On  ^  00 

CO  CMO  C?\J>»  O 

t^CMt^OOtN.  NO 

TfNOOO  ot^  o 

00  00  O  "^  O  00 

CONOmcoCM  O 


00    00000    00000  O 


ONr>>,  ocoi^ 
00  CA  cot^  »-• 

oiocooooo 

1^  Tf  CO  CM  CM 
OCM  NO  CM  O 
CM  ^  000 
VO  '^  CO  CM  '-< 

10  10  10  LO  10 


'-•VO  CO  ON  CM 

NO  — •CA'^CM 

r^  coot^  00 

CMCM  '-lOO-^ 

o  c^^  o  '-'  ON 

O  CAOO  r^  NO 
vO^Tj-rj-Tl- 


oooo-o   00000 


CM  "-I  CO  vo^i- 
rl-  (N)  ^  1^  r^ 

oo-^ooo 

ON'-^  OMDO 

000  cor^T^ 

CM  "^  10  NO  00 
vOTt  COCM  '-I 

OOCJCDO 


OCOVO^'':*- 
OOt^co  vo  Tf 

NO  CO  00  r^  CO 
On  vo  NO  c  >  10 
r-i  T-H  cvi  vo  c:^ 

OCM^NOOO 
-— I  O  On  lO  l^ 
■^  -^  t-O  CO  CO 

c5ocDoo 


00  vo 

00  CM 

CM  O 
v/o  v/^ 

^   NO 
t^  CO 

CO  CO 

d>d> 


r^ONr-H 

CM  NO  NO 
CMOt^ 
COLO  CM 

r^oo 

Tf  NDO 
Ot^io 
CO  CM  CM 


--I  COO  CM  vo  t^ 

r>sr^cN  -^Lo  ON 

tN.  C7N  Tf  r- c  On  cm 

LO  Ot^  NO  On  CO 

rf  '--1  r^  CM  CO  O 

NO  10  LO  CO  CM  CO 

CM  O  00  NO  vo  CO 

CMCM  '-"-"-I  ,-. 


000   00000  O 


ThNOt^cOON  '-•COCMCMO  OO'^OONOCh  Q  O  O  00  rj-  ro  O  vo  NO  O 

NOr-iTfvoco  r-HOOt^OO  OnCOtTCOOn  "^-t^vOTj-co  OOt^CMvot^ 

t-^r^CMOOO  cO'-i'^Ort  OCOcoconO  O^OOCOOn  COOnOCTnno 

rrCMOOr-iO  r^OCTNTf^  OOnONOO  O^nOvOI^  cmcmcococo 

C^JOON^Tl-  COloCMCMco  NOOvocm^  OCM'^OOC'-J  O'— i'-'J^CO 

Tt^cOTt-Tj-  -rt-LONOl^OO  CTnt- iCMrfNO  COOCM'^f^  OvococoNO 

ONCOr^NOLO  ■<:fcoCM'— lO  C3NC7\OOt^O  lolo-^coCvJ  CMCOvoC^lC^ 

iOLOlOlOlo  vovovololo  -tJ- -^  tT  rf  ■^  "^ -"^  "^ '^ '^  Tj- CO  CO  CO  CM 

OOOCDO  OCDOOO  OC3CDOO  O  O  O  CJ  O  CJ  C)  CD  CO  CD 


TfrfCMCMOO  CM 

'-"-I   Tj-  00  r-i  (N^ 

ON'-^CMNO'-i  n- 

'-^OVOTfr-^  CM 

CM  NOCO  COrl-  rf 

CVl  C?\  00  On  CM  NO 

t^rl-CM  OON  t^ 

CM  CMCMCM  T-H  ^ 

CD  (DO  CD  O  CD 


CO^CM  OOtv. 

Tj-  00  CTn  VO  o 
CM  l^  CM  LO  -^ 

N0O0n-hi>^ 
:-v  CO  On  GO  r^ 
OnOO  "-i  CM 
CO  CO  CM  ^  O 

NO  vo  NO  NO  ^ 

CDOOCDCD 


00  Tt  ON  CO  NO 

o  i^  o  01  c^a 

NO  Oi  CO  rr  .— I 

NO  00  T}- CM  CO 
00  O'^  CTnlO 
COLO  NO  t^  ON 

On  00  r^  NOLO 

LO  vo  vo  LO  LO 


CM  onlo  cor^ 

CO  LO  CM  LO  NO 

CM  to  c:^  .-I  O 

NO  '-'00  OOC?N 

CM  —lOr-H  CO 
>— I  COLOl>*  ON 
vo  -rf  CO  CM  '-' 
vo  vovovovo 


00000    00000 


•*1-VOCMO  vo 

OnnO  -— '  t>.t^ 
Tt  (M  CM  r-H  On 
"-I  to  O  NO  CM 

l-^  ^  l^  CO  '-' 

1-1  Tl- NO  ON  CM 
»— OONCOOO 
vo  O  -^  Tf  rt 

OOCDOO 


00  O  I^v  '-<  CVI 
NO  O  C?nCM  ON 
Tj-  COvo  CO  NO 
OCM  CjNCOt^ 

OOnCMOnnO 

vo  o  c^  chCM 
t^  Tt-  or^  vo 

Tf  -^Tj-CO  CO 

ooc>oo 


ONvor^ 

CTn'-"-' 
vo  O  ON 

r^  o^.  oc. 
cooo  o 

t^  COCM 

CM  C'CO 
CO  CO  CM 
CD  CD  CD 


OOOn  rt 

'-'  On  "^i- 

CM  NO  CTn 

VONO  C^J 

00  O  NO 

'— I  CO  vo 

NOTf  CM 

CM  CM  CM 

OC>  CD 


nOOOOOnCM 

00  coTJ■r^Tt 
oooo  oooooN 

CO  CO  CANOrt- 
OO  O  --I  COLO 

CO  00 1^  NO  LO 

NO  NO  nO  no  no 
OOOCDCD 


I^NOCM 

r^  ^  CM 

'-'  C7NLO 

00— ' 

■^rf  LO 
t^  CTn--^ 
■<^  coco 

NO  NO  NO 


COO 

r>»io 

00  CO 

t^O 
CO  NO 
CM^ 

NO  NO 


Tj- NO  CM  NO  NO 
CO  OLOio  O 

CO  CM  CO  NO  O 
1— '  O  CO  LO  C^l 
Tf  CA ''I-  T-H  ON 

COOCONOCO 


Ot^  On  On  On 
On  C3N  '-•  Tj- 1^ 
CM  CO  CM  NOLO 
t^  '— <  Tj-  10  vo 
t^t^t^  000 
'-"^t^OTt- 
t^  NO  vo  vo  TJ- 
Vi )  to  'O  to  vo 


00000     00000     00000 


vOCMOvovo 

OONNOt^O 

ONCOt^i\  ON 

C'l  !>.  NO  aj  CM 

CO  OnVOON  r-- 

t^  Ti"  Tj- 10  On 

CO  Ot^  rf  r-l 
VOVOTtTfTt- 

OCDOCDCD 


tv.ONNO 
OOt^CM 
t^NOrf 

oovor>. 
00'—  00 
coot^ 

CTNl^rf 
COCO  CO 

CDOd 


VO'-l  NO 

CM  C3N  CM 

Tf  vo  CO 

CM  CO  CO 

CTvCM  t^ 

NOt^  00 

CMO  00 

COCO  CM 

do  d 


CMvO'-*t^Tf 

CTn  ^  '-'  Oco 
CMt^  Tfcoco 
Cvi  t^  O  O  t>^ 
CJN  vo  CO  '-<  C?N 


Ovoor^t^  Tj-»-HCMCMNO 

CMOnOvconO  '-I --I  CJN  C?n  t}- 

TfT^r^coON  cOcoCOCJNtO 

>— '  CM  O  LO  NO  LO  O  >— '  ON -^ 

C^NChO-— 'CO  vo  O  T}- CO -^ 

LOOOCMLOOO  »— iLOOO'— 110 

O0NC?NC0r^  r^NOtoLO-'^ 

t^NOVONOvq  VONONONONO 

<Dci<z$ci<z>   (zic^dc^d  ddcDciiZ^ 


CM  Tj-  r-H  ,-t  r-t 

OiNOOTj-CM 
Tft^  COCDC7N 
to  CM  NO  NO  .— ' 
01^  TfCM  T-i 
C3NCMNOOTJ- 
COCOCMCM  »-H 
NO  NO  NO  VO  NO 


CVIOOCM 

OOONO 

0000  o 

COCM  rf 
OtOTj- 
00000 
O  t^  to 

NOLO  vo 


CM  NO 
ON  00 
cOTt 
co  '-' 
t^  CO 
coco 
CM  0\ 
vorf 


ONTj-'.tOO  r^ 

Tj-  CTn  CM  r-l  C-1  CM 

or^  CO  '-I  o  '-' 

CM  r-H  CM  ONt^  T-H 

'-''-'  CM  CO  vo  l^ 

"«t  »-  ONOOOQ  On 

r^  vTv  CM  o  00  NO 

■^■^  "^  Tf  CO  CO 


00000   00000   00000  o 


O— "CMco-"^    lONOl^OOOx     Oi-HCMCOTf    lO  vo  tv.  00  0\     OvoQ^OO    vo  ' 
COCOfOCOCO     COCOCOCOCO     Tt  rj- Tj- Tj- Tj-     tJ- tJ- t^  tJ- rj-     vo  10  NO  NO  t>^     t>»  ( 


110  0*0 

I  00  ON  On 


342 


TABLES 


H 
tn 
W 

w 

H 

p 
o 

<  U 


w 

W 
PL, 


/--S 

^ 


CO 
000 


fOTfOOVO 

r^rr^i>Hvo 

00  CONOCO  NO 

NO  00  CM  On '-' 

COOONOlO 

COCOLO-^l-Tf 

r^Ti-cMvo 

^iOt-h  COTJ- 

ooot^ogco 

t^LOcoOON 

OCMTft^O 

r>.Tf^CNjio 

NOQON-Hl^ 

ooOr^ONO 

VOvOONfO 

lonOtJ-co  CO 

rl-LOiOt^  00 

rOCh^  O 

vo  NOLO  r-H  On 

On  00  NO  CO  O 

NO  ^  NO  Tf  T-H 

OLOOONt^ 
00-^vot^ON 

ON  -^  NO  COLO 

On  00  CO  NOLO 

^ 

cnonotj-oo 

CO  l^  On  00  CO 

CM  NO  CO  COLO 

coa\o\c\) 

l>^  -rl-  lO  t^  .-1 

00  NO  NO  00  Ol 

t^co-HOO 

»-i  Tt  rv.  1-1  NO 

CM  ONl^LOTf 

CO  r-H  O  OnCO 

NO 

T^00^^O^ 

Tf  ONOCvi  On 

LOCNJ  OnNOtI- 

»-H  On  l^  i-O  CO 

»-•  ONt^NO  Tf 

On  00  00  t>. 

t^  r^  NO  NOLO 

lO  LO  -^  Tj-  rt- 

Tf  coco  coco 

CO  CM  CM  CM  CM 

CM  CM  CM  '-H  T-I 

--^OCJOO 

d>d>d>d)d> 

d>d>d>d>d> 

c5d>d)-d>d> 

c>»d>d>d>d> 

d>d>d)d)d> 

loooor^ 

rxOcoNOCN 

LO  On  Cvj  LO  lo 

OCM  On  LO  NO 

oONor^T-^^ 

t^  CO  CM ''t  CM 
t^t^  CO  NOco 

ctn  rr  voT^ 

^  rJ-COCOON 

<N  CNlTfCOON 

»-H  LO  VO  NO  On 

t4- CO  00  CO  ON 

OONt^sOvl 

NO  LOT-"  ON  00 

CO  ON  l^  r-ll^ 

r>»  r—\  \C)  o  CO 

C?N  CM  On  i-H  l>. 

C^l  O  00  CO  NO 
OTtrrON-^ 
coraoo  O  C3N 

OOCNJ  coo 

Cvj  •-«  OOfO  Q 
LOCNJ  NO  CGNO 

r-H>,  COCNNO 

»— <  »— •  On  Cv)  CO 

OOTf  tM^nO 

^ 

coooot^ 

On  NO  CO  CO  O 

O  T-H  C-l  LO  t^ 

00  ON  00  LOO 

vooO'-^Loo 

CM  rN.  coc^ 

S^^S^:^ 

CO-<*  NOOLO 

1-t  COLO  coo 

»-•  00  NO  -O  LO 

lOi-Ht^LoCM 

lO 

lOO  ^  0^ 

OOlocO'^  ON 

t^LO  -^CM  «-• 

CO  CO  CO  CO  CO 

On  00  NOLO  Ti- 

On  ON  00  00 

t^  l^  t^  NO  NO 

vo  LO  LO  LO  LO 

Tf  rf  Tf  rf  CO 

CM  CM  CM  CM  CM 

'-;  o  o  d  o 

d>d>c5d>d> 

d>d>d>d>d> 

d>d>d>d>d> 

d>d>d>d>d> 

d>d>d>d><zi 

vO 

8^§;^ 

LOTtNOCOfO 

00ThNO'«t^       T}-CN10Nt^0\ 

NOt^OONOOO     Tj-cococot^ 

VOCOONCOt^ 

QOOt^  ON  CM 
NO -^  CO  NO  O 

©=^ 

r>s  o\^ '-' 

r-H  U-)  00  LO  -rj- 

r^ooco^CM 

OONNOO--H 

C^t^OOco 

cooo^t^^ 

^ 

COCVJ  0\V0 

LO  OnCNJ  00  o 

n-oooo^ON 

CM  ONNor^i^ 

(MNOt^OO 
t^rf^f-^co 
vOTfco  1.  1  CO 

'^OOOLOO 

Nor^t^  cot^ 

QNt^CvlvO 

On  ^  NO  CM  On 

NO  NOLO  to  LO 

t^Tf  NOLOO 
CMOOTfr-HON 

T^ 

VOJ^VOOO 

cgr^'^corvj 

rr  NOONCOt^ 

lO'-tr^  CO 

O  NO  CO  Ot^ 

r-H  On  l">.  LO  CO 

T-H  ONt^NO  Tj" 

CO  »-i  O  ONtN. 

ON  On  00  00 

00  l^  I^  I^  NO 

LO  -^  ■<^'^  rt 

Tt*  CO  CO  CO  CO 

CO  CO  CO  CM  CM 

'-^cJddd 

d>d>d>d>d> 

d>d><5d><:^ 

d>d>d>d>d> 

<5d>d>d>d> 

(ocDdxzici 

vO^vOOn 

^CO^T-HTt- 

r^  CO  LOON  00 

OCOLOCMCq 

LOO  On  CO  tv. 
ON  NO  CO  CO  rt 

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rlLOOOCMr^ 

^ONOOO 

irj  ^  CM '-H  Tj- 

00C^^LOTtTt■ 

OO  ^  ^  '^ 

r^  Tj- 1^  o  NO 

"^  o  l^  '^  "^ 

NO  00  On  l^  t^ 

T^oocoooc^^ 

NO  COLONO  i-l 

COLO  0\  o 

<N-H^ON00 

NO  O  t^  C^l  rt 
CM  ON  CO  NO  NO 

00  COLO  CM  CM 

T-H  00  T-"  l^  LO 

-^NOooT^No 

LOONO  CO  o 

^^5. 

lOLo  OnOO 

On  CO  On  NOLO 

XOLOLOLO-^ 

CO  00  On  t^^ 

vocM  oom 

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(V)  ON  LO  COO 

LT  J  On  '^  O  l^ 

LO  coco  CO  rf 

NOCO  '-'LO  O 

Tf 

r^  Tf  CM  Ot^ 

LO  CO  '-'  On  t^ 

LOCOCM  O  On 

OS  0\  00  CO 

00  l^  !>.  t^  tv. 

NO  NO  NO  NOLO 

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CO  CO  CO  CO  CO 

'r^d>d>d>c5 

d><Dd>d><:5 

d>d>d>d>d> 

d>d>d>d>d> 

d>d>d>d>d> 

d>d>d>d><o 

tN.O'-HCO 

r>v^vovor^ 

»-<  .-lOLOON 

CM'-hOO'^On 

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COMPOUND    INTEREST;    OTHER     COMPUTATIONS 


343 


•-^  O)  Tf  c^i  in 

OTf  rv.  vo  1— I 

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t^  O  lO  t:!-  CO 


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t^CA^CM  Tt- 

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NO  NO  On  CM  CO 

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t^iovor^io 

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CO  CO  coco  CO 

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vo 

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CM 

to  Tl- CO  CM  '-H 

On  00  00  l^>.  NO 

--^OC^nOnOO 

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t^NOlO'<^CO 

CO 

CO  CO  CO  CO  CO 

CM  CM  CM  CM  CM 

CM  CM  CM  CM  CM 

CM  CM  '-  ^  r-H 

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t^t^cocor^ 

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CM 

00  CO  CO  CM  T}- 

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COCOIO  CJNtO 

ONOCO»-H  ON 

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CTn  O  CD  C3n  CO 

CO 

ONCNCOO  O 

coo  ON  CM  t^ 

lONOCAu^ro 

Tt-t^CM  OnOn 

i-^r^t^^CM 

00  NO  ON  NO  NO 

^ 

lo  lo  •^lom 

NOt^OOOCM 

T^-NOCTN^Tt 

CM 

i-H  onoo t^^ 

lO  TfCOCM  '-' 

OONOOOOt^ 
CO  CM  CM  CM  CM 

NOlOTl-Tj-CO 

CM  ON  NO  rf  CM 

\r> 

Tj-  CO  CO  CO  CO 

CO  CO  CO  CO  CO 

CM  CM  CM  CM  CM 

CM^^^^ 

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ddddd 

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d 

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1— c  CMlOlO'-H 

or^i^-^CM 

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Tf  CO  r-H  1-^  CO 

CO  t^  00  ON  CO 

C^gg^g 

t^  1— it^  On  '-H 

t^mNor^t^ 

CM  rf  NO  CM  00 

NO 

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cor^  CAoOTf 

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t^  Ot^^  '-• 

•^ 

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ON  lO  -rj- vo  '-^ 

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CO 

CO  •-<  On  00  t^ 

NO  NONo^r^ 

1^00  O  "-I  CO 

rfr^  C7\'-<  Tf 

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NO 

T^  CO  1— '  o  ON 

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On  00 1^  t^  NO 

m  CM  On  tv^  T:t- 

COT-HONOOt^ 

NO 

Tl-  Tt  Tt-  Tt  CO 

co  CO  CO  CO  c»^ 

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CM  CM  CM  CM  <N 

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cococococo    cococococo    '^  "^  rf  ■«i- rf    Tf  tJ- t^ -^  tj-    toioNO^t^.    In.' 


iOO»0 

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344 


TABLES 


CO    CVJfOioOO 

VO    lO  t^  1— t  CA  00 
lOto    oovooiooo 

CV>  CO  fO  t-i  T-i 

CO  l>%  -^  "-H  U-) 

lOOO  TfCOlO 
CVlCOlOt^ON 


CM  r-i 

lOiOCM 

OO'-H 


»-<C^cOTf   »ovot>.ooo\ 


t^^coONt^OO 
i-H\OCM  ^CM 
OOMDiO  Tj-io 

CO  "^LO""^  0\ 

CO  COIOO  CO 
OOOOntJ-^ 

Csi  "^  r^ '-' »J^ 
t-h'  cnJ  CO  lo  vd 


V0C0»O»-tC0 

VOTfOOOt^ 
CviCMcorJ-O 
0\CNU>.  CO  o 

COOO^O  OOrJ- 
C^coOOcoON 


VOCM 
■rj-  CO 

lO   «-H 

iot< 

CM  CM 


OOOO 
On  CO  C^ 

lOCM  CO 

OOtJ-O 
(M  ^  On 

com  CO 

odocM 

CM  CO  CO 


coco  CO "^r^ 

COOOO  OiO 
VOOQOO\ 
l^l^O  OOCM 

tN.  r-«CM  On^ 

in  y-<  l—^  U-)  in 

ripOxGOOO 
CO  CO  CO  CO  rj- 


Q 
o 

w 

u 

w 

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^     Q 
H     !^ 

w    o 

<: 

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CM 


CM 


^ 


CM  CO 

VOVO 

8  CO 
o 

CMOOt^ 
CMOCO 

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t^Tj-ooooo 

OM^CM  '-'  CO 
'-hOnCM^On 
»-i  t^  VO  '-H  0\ 
Ot^OONCO 
corfONiOiO 
CM  CO -^VO  00 

irJvdKocJOx 


rj-t^r^vooo 

OOCMt^t^  00 

o^CMr^o 

t^OvCM  CM  t^ 
iOTl-(M00co 

OCO^OOnCM 
t-H  CM  CO  Tj^  vd 


O'-Ht>X^V0 

COt-hioiO  VO 
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OiOnt-h  vOTf 
'— I  CO  O  t^  CO 
OniocoCMu-j 
OO  CO  ChOO 

vqp'^oqco 

t<C7NO  >-<  CO 
^^CMCMCM 


COCOON 
OCMCO 
OOnOn 
CMCMTf 

lOONO 

r-iCMr^ 

^t^vo 
O^T^o 


r-(00 

OCO 
CM  CO 

^^ 

ON  NO 

vOco 


vocoooot^ 

On  VO  li-)  T-H  VO 
VO  '-'  O  CVI  VO 
»-i  cor^  CMi^ 

COt^OTl-00 


OOO' 


CM 


cOTfVOOOO 
COCOCOCO  '(I- 


\OVOC50iOfO 
i-hOncoi"  ■ 
00    OOCO( 


§i2 


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■NO     O  "-I  (M  CTn  VO 

"^OOTj-CMTt 

(MMDCM 

iCMCO"^   iovdKooo\ 


OCMCO  CM  irj 

0"^I^IO  r-( 

1— I  lo  CTn  1— I  00 
CMt-hOOcoco 
t^t^Oco  On 
On  00  CM  OCO 


OOsJf^"*"^ 


CMiOMDOOcO 
C7NCM  On  covO 
VOLOOCM  00 

'— I  oor^  T—i  lo 

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CO  ON  CM  CM  O 
OnCO^  ^  Tj- 
CM  vOO->4-00 

K  00  o  '-H  cm' 

i-H^CMCMCM 


OOn  Tt*  »— I  t^ 
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coco  On  0\  00 


r^coooTf  r-i 

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CM  i^  CM  CO  Tl- 


CMCMCOt-h  T-i 

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ON"^  CO  OTf 
On  OCM  T-(  CO 
CM  ON  CO  CM  CM 

CO  t^rt  lO  0\ 

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CMcouSr^cd 

CO  CO  CO  CO  CO 


CM  CO 

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lOOOCM 

t^CM  VO 

»-HlO  O 

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CO  On  CO  Tj- CM 
CMOOOOCM- 
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t^  vot^  o»o 
T-j  CM  CO  lo  vq 
tovdt>Ioc)ON 


tJ-On 
On  CO 

^^ 

CM  ^ 
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CJCM 


T-HCOCO 

t^o»o 

CO  CO  CM 
OTfco 

r-(lOlO 

LO  VO  ON 

CMiOO 

CMTtr>. 

corj^io 


VO.-IVOC0  00 
coCMiOVOt^ 

ONr^  VO  "^lo 

Tft^QCO  VO 

■^  VO  'O  VO  T-1 

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On  CM  VO  On  CO 
vd  00  ON  O  CM* 
,-1  ,-H  ,-•  CNJ  CM 


ONOOOor>. 

T-i  COCM  C7NCO 
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•-"OOCMio  T-i 
VO  CO  On  VOO 
^-H  VOlo  O  •-< 
O  T- 1  u-j  CM  '— ' 
t^y-nnCDm 

CO  lo  vo'  00  On 
CM  CM  CM  CM  CM 


lOONOOt^tN. 

T- 1  vocoi^  VO 
C?NOnCM  CTnCM 


t^  O  Ot^  CO 
CM  t^  Tl-  ro  VD 

O  LO  '-I  Tn.  CO 

»-J  (Vi  r!^  LO  t< 
CO  CO  CO  CO  CO 


00  COCOON  r-iC3N 

CO  C?N  Ov  "-I  '-'  VO 

irjco  VO  O  -^  On  '-t 
VO                   CMOVOloOncoco 

©^                  CM  On  CMloOnOOco 

vc<             roLOO  (MOnCMCMON 

kJs,                   l-HTl-ON  lOCMCMCOLO 

OOO  riCMco^iO 

JcMfO-^  irjvdt<odO\ 


1>.OncoOlo 
vOTfTfvoO 

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t^  CM  CM  00  CO 
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t>.  0C3  O  CM  ^ 
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00"^OnCMVO 
t^  00  CO  l^  CO 

r^  ONLOLo  VO 

CO  VOLOt^  y-t 
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VOOnCM  Tft-^ 


CiS'-i'^Tl-O 

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r^  CM  On  CO  O 
VOCM  t^Tj-CM 

VO  LO  LO  1— I  LO 

CO  Of^LO  CO 
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T-H  Tf  OOCM  VO 

coTfiot<od 

CM  CM  CM  CM  CM 


'-Hvoor^to 

NO  OnlovOOO 
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CM  voiN.r^  o 

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CO  CO  VO  '-^  00 

VO'-iOOCOCJN 

CqiOONTj-ON 

C>  t-J  CM  -^  LO 
CO  CO  CO  CO  CO 


tOiO 

CM  ON 

VO  VO 

iHvovo 

Sjt^LO 

Scot^ 


CO"* 


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0"<^CMC7nO 

CM^VOO(M 

CM-^r^oo^t 
r^  LOCO  001^ 

LO^OOCO  CO 

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CM  ONVOlo  VO 
^t-hCMCO-^ 
irjvdKodON 


t>sOCM"*00 

CO  CM  Tf  On  00 

NO  t^  »— I  lO  ON 

VO  CO  VO  '-I  t^ 
VO  On  CO  1— I  CO 
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LOt^  COO  '-H 
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COMPOUND    INTEREST;    OTHER     COMPUTATIONS 


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346 


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to  t^  CO  000 

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0 

348 


TABLES 


VOIOVO'-" 

ovoor^co 

co^Ot^t>^ 

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^ 

ONTTCOVf 

ooio  orv^io 

CO  00"^'^  CO 

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OCNlCNt^ 

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t^  NOLO  coo 

NO  ^  NO  ON  CO 

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COMPOUND    INTEREST;    OTHER     COMPUTATIONS 


349 


CM  Ti-  .-H  oor^ 
o  >^o  o\ '— I  CO 

ON  CO  00  eg  r^ 

O  ^'  -h"  (VJ  (NJ 


COO'-'  o-^ 
t^t-H  OOco-^ 

.-'Ol  COVOCO 

lO  vot^ooo 

Tj- lO  >r)  Tl-  CO 
'— '1^  On  cot^ 


lOOcO  Omo 
OOC»MDtJ- 
lOCM  nO^^On 
l>.  CM  O  -^  ■'^j- 

r^  .-I  NO  "^00 

CM  NO  O  ^  CO 
ONOCMNOO 
,-, -rf  CO  T-*  xn 
u-j  to  trj  NO  Nc5 
CM  CM  CM  CM  CM 


VOCMiO^-it^ 

OONOvot^T^ 
CO  C3n  CM  COON 
CMNOCX)iONO 

0.-H  Tt'-l  CO 

C0'<^t^c0'-' 

COLONO  t^t^ 


OOOOCMOOO 

NOCMTfOON 
T-H  ONNOt^NO 

CO  On  NO  IT)  00 
CM  CO  00"^  t^ 

NO '-H  ONOC^N 

cOt^C7\ON00 
00  On  c5  ^  cm' 
CM  CM  CO  CO  CO 


TfCMVOOOON  NO 

"^tCMOOTfOO  CM 

Ot^TfOO  O  in 

Tt  r-Hr-H  «Ot^  O 

r^oocMt^'-'  1-H 

CM  T-i  NOm  On  rl" 

CM  voOnNONO  t-H 

r>.  Tt  p  NO  T-<^  NO 

COTj^lOU-JNO  NO 

COfOCOCOCO  CO 


in  NO  On  t^  00 
COOO'-'t^'-' 
On  r-i  r^ONNO 
CM  CM  r^  CM  CM 

coo  NOLO  00 

u-jr^t^  t^  NO 

Tj- -"^  CO  1— I  00 

Vq  T-H  NO  T-H  U-3 

'-H  CM  CM*  CO  CO 
CM  CM  CM  CM  CM 


t^C?\00  00NO 
irjiOOO  '-I  NO 

ONNO»J^O\t^ 

l^  NO  NO  On  00 
lOTfcoCM  CM 
Tt  On  CO  NO  00 
O'^ONCOt^ 


^TfOVOON 
CMt>^C7N^NO 

CM  r-tCOCOON 

CMCOOCOO 
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NONONCJKt^ 

CM  CM  CM  CM  CM 


OO0Ntv»0\ 
inirjLot^  ,-t 

a\rf  CM  t^o 

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lOTj-lOONlO 

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T-(  Tj- 00 '-I  lo 


t^OOiOvoCM 

CMcooOQOn 
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cor^Ti-00  00 
00  CO  t^N.  On  p 
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li-)  ,— I  rl-  CO  »— I 

lO  Ocovo  0> 

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cm'  CM  CO  CO  "^ 
CM  CM  CM  CM  CM 


COOO  '-"  Oco 
cOTfrfNOOO 

c^cM  coooo 


OOOQOnOCM 

c^oONO'^o 

On  Tj- On  Tj^  On 

•^XOlONONO 
CM  CM  CM  CM  CM 


rtmoocor^ 
CM -^u-)  coo 
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t^  00  On  NO  NO 

lOONTl-r-i  On 

»J-J  ONCONOt^ 

cot^CM  vo  O 


t^QVOl^»-< 
OOOOnioo 
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OCMNOCOCM 
On  O\00l^"^ 
TfOOCMNOO 


ONCMOOmON 
OOiONOiOCM 
to  t^  NO  lO  On 
OOOOONOt-h 
NO  t^  00"^  NO 

COTj-Ot^CX) 

CMr^NOC7\C7\ 

'^^  T-H  W  ,-H  Tf 

'-I  corf  NO tv! 
CO  CO  CO  CO  CO 


COO\C?N 
com  CA 

TfCOON 

1-H^OO 

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t^Tt  ,-H 
NOt^t>. 

odo\CJ 
cocoTt 


NOTf  Tt 

T— iLO  NO 

C?\CM  T-H 

CM  CM  IT) 

On  O  CO 

NCO  00 

00  00  On 

mco  O 

t-h'cM  CO 


<«1- O  t^  T-H  CTn 
rO»0  Onu-)00 

C?\NO00C?Nt^ 

■<^t^VONOt^ 

00  00  00  ON  CO 
lOONCOl^CM 
CO  'O  Tt  O  NO 
1-H  t^  CO  CA  Tj- 

CO  CO  Tf  ^'  vo 
CM  CM  CM  CM  CM 


OCOtOt^t^ 

ooomtoLo 

T-HCMTl-T^OO 
tOiOrJ-OOCM 

CM  t^  QCM  NO 

t-^CM  0\NOrt- 


Si 


NONOt^t^OO 

CM  CM  CM  CM  CM 


ioC?nnOCMt-i 
lO  00  CO  CM  CM 
CNt^i-HtoCM 
CMcoOMDt^ 


OCOnOnOO 
CM  CM  CM  CO  CO 


T-H  T-H  T-H  On  ro 
NO  CO  00  CM  NO 
CMTfCMT-H  o 
NOt^TfOOO 
CM  Tf  On  00^ 

NO  NO  00  CO  T-H 
NO  T-H  lO  ON  CM 
0^ '<:*•  00  CM  t^ 
O  r-J  ,-H*  CM  CM 

CO  CO  CO  CO  CO 


VOOCMOOl^ 
rj-mmt^NO 

ONIOIONOCM 

OTf  oo<?\0 

CM  "^CMOOn 
T-HirjcoOt^ 

Tj-CONO'^t^ 

^T-HChNOT-H 

roiONOodo 
CO  CO  CO  CO"* 


t-HTj-ONt^lO  Tf 

t>vt^t^CONO  O 

t^'^'^OCM  CO 

t^coOT-HCo  t>x 

-^OnOVOOn  Tl- 

t^  CTmo  T— I  tN.  ,_ 

OOt^NOlOTj-  NO 

lo  00  p  1-;  T-H  o 

t-h'  CM  rf  lo  NO  »< 

Tf  "t  rt  Tj- Tf  Tj- 


T-H  CM  Tt  CM  ON 
OGOI^rfTj- 
00  vo  00"*  00 
CO  "*  covo  CM 

00  T-H  ^  ONt^ 

lo  Nor>>  00  T-H 

T— <  r:J-  NO  r^  00 
p  NO  CM  00  Tf 

Tt  Tt  lOVONO 

CM  CM  CM  CM  CM 


OOT-tOcoOO 
lOCO"*  VOOO 
r^r^r^^CM 

OnOOCMvoOO 
vo  NO  "-H  O  vo 

VOOt^  VOTf 

t^  NO  CO  O  NO 

ONOCNOO  CO 

tvlt^ododoN 

CM  CM  CM  CM  CM 


OOntM^CO 
CMt^OvoCM 

voOOt-hCM 
Tt  NOVO  CO  CM 

OOONOCMNO 

VOOOTfi-HO 


OOcoioioOn 
T-H  00  On  NO  CO 

t^  ON  T-H  CO  CO 

COOO  OvvoOO 
CO  Tj-  T-H  vo  NO 
CM  NO  CO  CM  Tl- 
lovovorf  CM 
lO  Ovrjo  vo 
CM*  CO  CO  Tt^  ■'^ 
CO  CO  CO  CO  CO 


t^r;^c7N00co 

opooo^oo 

CO  ,^  00  NO  T-H 
OO^NOOOt^ 

vo'^CMrv.oo 
Chj-^Ot^'^ 

On  CNj  crj  CO  T-H 

rt  t^  C3N  T-H*  CO 

co^^coTf* 


■«tO\00 
COrl-T-H 

8  CO  CM 
CM  CM 
NOcot^ 


cmtj-   r^ 

vo  vo     NO 


5^ 


TfO 

vno 

CO 

r-) 

r^ 

g^ 

;^ 

CM  Tj- 

vo 

ONO 

,_! 

Tl-VO 

lO 

lOTl- 

NO 

COTt 

CO 

Ot^ 
NO  CM 

0\ 

CO 

CO 

NO  T-H 

S:^ 

o 

coOootOTh 

CM   ^T-H   '-HTj- 

nOOtJ-CM  NO 

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ON  CM  CM  ON  Tf 
OOOn^tI-O 
OONOrfONO 
(50  vOCM  ONvrj 

Tj- vo  NO  M3t>I 
CMCMCNlCMCM 


CMt^CTNCOCO 
CM  CO  vo  CO  00 
00  t^  CM  T-H  On 
vo  NOOOO  T-H 
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t>^t^  OOCM  00 

OTfr^O'-H 

CM  00  Tj- ,-H  t^ 

OCJodONOO 
CM  CM  CM  CO  CO 


NOCMrJ-ONO 

T-H  CM  rococo 

CO  vo  oco  vo 

CO  CO  T-H  t^  ON 

On  00  CO  ■*  CO 
NO  t^  T-H  t^  NO 
CMCM  CM  OOO 
CO  On  vo  >-H  NO 
T-H  '-H*  CM*  CO  CO 
CO  CO  CO  CO  CO 


»OCMCMt-hO 
CMCM  -^ONt^ 

OOCMTj-ONO 
NO  00  CM  00  CO 
T-H  OONO-^vo 
OOCM  O^vo 
VOCM  00  cot^ 
CM  00  CO  On  Tf 

Tf  Ti^  vrj  vn  NO 
coco  CO  coco 


Tff^NONCMCNj 

t^  NO  t^  CM  NO 
vo  NOt-h  ovo 
t^  00  ON  00  t^ 
00  MDvo  On  NO 
^^^IZJ'^OCJN 
T-H  p  CO  CM  NO 

pvqocoTf 

InI  C?N  CM  Tf  NO 

CO<*^TfTtTl- 


r^  NO  001 

CM  Ovo' 
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t^  VOVO' 

CTnNOO' 

rf  CO  T-H  00  Tf    C7\ 
00  O  CM*  CO  lo     NO 

TflOlOlOlO     lO 


CMt^r^toON 

CM  CO-*  vo  00 
00"^  On  CM  vo 

OOO  00  ON  NO 
t^CM  VONONO 
t^  CM  On  On  CM 
O*N0  00O 

00  vo  CM  ov  ^>: 
toNot<i^od 

CM  CM  CM  CM  CM 


{Tn^-^-OOO 

SOCTncoOn 
vo  On  CO  CM 
00  O  ONO  CO 
vo  vo  VONO  O 

00  t^  On*  CO 
OOOnOOno 
*-Ht^rl-^ 

On  O  O  T-H*  CM 
COCOCOCO 


i-HCM-^NOCM 
T-HCM  i-H  Ovo 

NO  On  00  00  CO 
00  00  OOvo 
nOn3t-hO* 
Tf  OnOOOvo 
CO  ON  vot-h  vo 

OO^-r-HOO* 


CM 


TfOOONO^ 

*OOTft^ 

00  NO  On  On  00 
O  CO  On  vo  t^ 
voCMNOOnO 
■*t^  CO  CO  00 
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Ot^COC^NVO 

NO  NO  t»i  ^s.■  00 

CO  CO  CO  CO  CO 


COVOr-Ht^lO 

»or-;"*t^co 

I^CMoOt^* 

T-H  On  CO  O  T-H 
T-H  T-H  ovo  vo 
NO  fi  VONO  00 
(?\*vocM  NO 

T-H   T-H   C7N  NO   T-H 

C^NCMrttviO 

CO**-*  in 


O'-'C^cO"*    lONOtxOOOx    Q'-hCMcO*    iOV0tN,00O\     0»00i00    lOOtOOiO     o 
fococococo    COCOCOCOCO    '*'*•*■*'*     *"*■*"*"*    iOvovOVOt>.    t>.  00  00  On  CTn     O 


350 


TABLES 


Q 
O 

Ph 

w 

u 

o 

Q 
"A 

<   < 

Ph   ^ 

^  S 

W        l-H 

H   ^ 
< 

o 

W 

O 


W 


00 
CO 
009 


CO  tV.lO  1— < 

ON  CO  TfT-Ht^ 

looorfvoco 

ONt>.ON00C3N 

CM  CM  CM  00  CO 

VO  ON  Tj-  00  rj 

CM  voOnvo 

t^  CO  Tf  00  CVl 

CDiOOvON  0\ 

c^cM  vo-*"^ 

OOuoOncovo 

CM  VOIRON  to 

»-H    ,-H    ^  (Nl   O 

VOCM  '-IVO 

CO  Tj-  r-H  CO  CM 

t^-^  CO  CM  CO 

— *  VO  i-H  00  t^ 

vovOrt-rf>-H 

0\0n— <o 

vo(^^ooc:^ON 

OOt^rtcOOO 

'^CMOO'-i 

CM  IN.  00  t^  to 

to  voco  VOCM 

^ 

cOfOO'-' 

cococor>N  VO 

OCOOOVOON 
O  vOcoCM  Tt- 
VOOOOO»JO  On 

CM  00  CM  VO  l-H 

OOtOCOCO 

CO^-i  tOT-il^ 

CO  CO  com 

CMt^CM  0>  r-i 

CMiOI^t-xOO 

ONTfr-H  CO  O 

S3S28^ 

VO 

Tt-cor>xVO 

--ht-hOOOO 

'-I  Ot^CMtO 

VO  VO-*  Oto 

ON00VO-=t 

C^l  On  Lo  CM  00 

CO  00  CO  00  CM 

t^—'TfOO'-i 

tM>vO  coto 

t^  p  CM  ""^  to 

Ot-h'cMCO 

rt  TtiOVOvd 

KKociodcK 

OnOOCDi-; 

»-5  T-H  cm'  cm  cm' 

CM  CO  CO  CO  CO 

lOCOCO  O 

r^vo  o  vooo 

CO  CM-*  ON  Tf 

TtVOiOOVO 

Th^OOOOCTN 

tv.OCMVOOO 

ON'^OiO 

vOO-^t^*  VO 

ON  CVl  VO  C3N  On 

O  i-O  <M  ON  00 

OOCTNVOVOO 

CO  r^  to  00  IN. 

to  CO  VO  CM  tr) 

ooooo 

VOCM  roCM  ^ 

'^l-M-'-'CNO 

O  CM  CVI  CO  '-H 

TttOCOt^CO 

in 

OOi-HTfLO 

r^ONi^ '-'  CM 

CO'-HtOt^-"^ 

irjVO  VOOOCM 

CM  '-'Otovo 

■^OOCOCMt^ 

COtJ-CMOn 

TfvocoCMOO 

^^T;^CMlo  VO 

VOt^OiOCO 

ON-HOr-HO 

CM  ONCOiO 

CMO  VOCOI>^ 

T-H  VOCOCOC50 

ONt>v^CMO 

CM -H  CO  00  00 

CO  to  CO  00'-' 

iOU-)CM  Tj- 

CMt-xOO  VOO 

CM  O  VO  OOn 

tN.  cot^OOOO 

VOCM  VOOOOi 

CM>.  Tf  C?N  ■* 

0\00t^>0 

COOt>^Tfr^ 

t^  CO  00  CO  00 

COOOCM  VOO 

't^OO'-H'^I^ 

OCOVOOO'-H 

O^'CMCO 

TtlOlTJvdK 

KOCJOCJONON 

O  O  t-I  i-h'  CM 

CM*  CM  CO  CO  CO 

rf  Tf  rt^  Tf  to 

Ovomo 

-■^oo^t^o 

00  CM  00  CM  00 

COIOCOOON 

tOOO  VOCDnI^ 

VO  to  CV1 '-'  CO 

^ 

oorN.coi^ 

t^  -^  C7N  O  VO 

-HONt^rl-(N 

t^O^OOiO 

rf  00  1^00  CO 

CTNTf  OOtoto 

tN.r^rl-iO 

VOOJOVOO 

OOVOOCMiO 

U-5  to  .— 1  T— 1  CO 

VOCO^'^OO 

00  '— '  CM  CO  00 

:^ 

COVOVOCM 

l^  t^  O  CO  0^ 

r-H  ,-1  ooiocg 

;?;s2^8^^ 

COCM  CMl^tN 

Or-HOt^OO 

CM  voco  00  00 

On  VOOMO 

o\ooi^  cor^ 

rv.  On  to  00  00 

OM^  ■*  tv.  Tj- 

Tj- 

vocMoor^ 

On  t^  C^4  lO  00 

CM  00  00  CM  CM 

CATft^ONCO 

t^Tl-Tft^lO 

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OOLO  On  On  VO 

CO  CO  O  to  On 

OO00T^C7^ 

C^Tj-tOTf  CM 

C3N00t^iO 

CO  .-;  00  "^  CM 

ONio^^vqcM 

t^CMt^  .-H  to 

O'^t^^'^:!- 

00  i-i  rt  t>.  p 

CDi-h'cMCO 

T}-'  Lo  liS  vd  tN." 

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352 


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CM  "^00  NO  lO 

CM  NO  CO  On  NO 

»— '  "^NOt^lO 

Ont-(U-)00  00 
'-'OO  NO  NO  00 
CM  ONf^mco 
rf  CO  CO  CO  CO 

ooooo 


CM  OON '-•^ 

m  Not^mo 

ONCMO'-''"^ 

cmoonoon: 
CM  t^cM  ONr~ 

CM  00\WNC 
CO  CO  CM  CM  C^ 

ooooc 


ooooo   ooooc 


CM  NO  00 

ONONt^ 

t^CMTf 

t^oo-^ 

CMCOTJ- 
NO  00^ 
ONCM'^ 
TfcoCM 

id  do 


CM -H  NO  (N  CM 
COCM  '-'  ooo 
On  LO  NO  T}-  On 
OOCMmOOO 

CO  lOLOlO  NO 

'^ONOOOTf 

Onno  cO'-i  O 


00  Tt  On  NO  CM 
»—•  00  On  coco 
Tf-  coOnOco 
CO  On  t^*-^  CM 
rl-CM  NOCM  t^ 
cOTfNOO'^ 


OOOOO     OOOOO 


NO  CO 
COO 

rt-NO 
CJNt^ 
CMO 
vom 

OO 


Nooot^ 

NOt^^'^f 
ONIJ^OO 

t^  ot^ 

ococo 
CM  oom 

OOO 


rro 

lOtO 

n-No 

CM  00 

COO 

OO 


COt^    T— I 

coOrJ- 
OCOCM 

OO^'^ 
CO  CO  CO 
OOO 


lO  NO  I\  00  Oj 
"^  On  CM  O  M 

CO  .— t  IT)  r-l  0> 

NO  CO  '-I  O  bi 
CM  t^  coOb 
co^  OONb 
CO  CO  CO  CM  C 

OOOOC 


.-•t^CM  ^'-iCM'^iO 

rJ-'-«0  •-•  CO  r^v-*  ID 

Tf'-i'-i  CMcoOOcoO 

OO^  NOcoOOcot^ 

OOCMCO  OiOiONONO 

VOCTnlo  uO'— tt^ONLO 

ONCM-^  CJNNOcO'-iO 

TfCOCM  ^r-lT-H,-!,-! 


ococo  _ 

CO  00  00  T— iiO 
ONOiOCM  O 

to  CO  r^  CO  00 
■^  LO  r>»  T-H  IT) 

ONCOt^t^NO 

ooooo 


NO  CM  CO  On  00 
Tj-r^CM  t^rr 

NO  NO  O  ■'^LO 

CM  '^noOOlo 
ooo  T-toO  On 

,-H  NO  COCTNNO 
NOLOLO  Tj-  tT 

OOOOO 


On  00  00  NO  lO 

CO -^  CO  NO  NO 

or^ CM  NO  NO 

CM  CO  t^  On  00 
CO  ONt^r^  On 

"Tf  »— '  ONt^LO 

rl"  -^  coco  CO 

OOOOO 


r^ONt^fOo 

■<*CMt>.VOC* 

CMivsNor 

CM  00  NO' 

cor^  coOo 

'^CM'-'OO 
CO  CO  CO  ""^ 

ooo< 
ddd<5c 


Tj-'-iO\  Ot^COONNO 

Tt '-<  O  OOCOCMCMCO 

CMCM-H  OnOOOOOO 

^  CM  00  co-^CMC^NTj- 

voOCM  OLONONOt^ 

t^ONO  NOCMOOONO 

Onco-"^  OiNOcoCMO 

rj-  CO  CM  ,-1   r-H   ,-4   »-H   ,— ! 

iddd  <od)<Z)d>d> 


00  00  ON  CM  l^ 

o  ot^oo »-" 
CM  T^  00 '^j- '-< 

COLO-'^'-'O 
LO  "^  00  -^  On 

LO  NO  00  CM  NO 

ONOor^t^NO 

OOOOO 

<Z)  cS  CD  <d>  d) 


OOONOlolo 
tN^NOOOt^ 
CO'^OOCM  r-H 
CM  -^moOLO 
'—1  ON  CM  On  O 
CMt^'^OCO 
nOlololo  rf 

ooooo 
c5d)<:6<od> 


'-iioCMTj-t^ 

cor^t^oorj- 

LOO  COLO  CO 

'—I  CO  NO  00  t^ 
""cfOOOCOO 
LO  coOCOt^ 
rl-  -rf  rf  CO  CO 

ooooo 


5i 


loooco  ,  ^ 
r^ooLo-^c 

Nooom^ij 

ONOTj-NC 
'^OOTfiFHC 
LO  COCM  '-'C 
CO  CO  cococ 
OOOOC 


mNOt^00C>>     OOCMCO"*     lONOt^COON     O'-iCMcOrJ-    iT)  no  t^  00  C 
^^^^^     ^^^^^     CMCMCMCMCM     CM  CM  04  CM  C 


COMPOUND   INTEREST;    OTHER    COMPUTATIONS 


353 


txON00O\M3 
t^  CO  ^iT)  O 
iv.  t^t^  00  o 

ooooo 

000C)0 


oooooc^in 
to  U-)  On  r-H  ,-( 

LO»— I  OOVO 

oor>.  vovoio 


covo  vooot^ 

CVJOOt^OOCO 

COVOOq  r-H  CO 

00  CM  t^  CM  r^ 
TfTfCOCOCNJ 


CVJVOON 

VOCMO 

^^^oO'<:^ 


On  00 

ON'* 
to  CO 
OCM 

r-to 


ooooo 
doodo 


ooooo 


ooo 


oo 
dd 


vOONOrOCVl 

O  '— '  "^  ^  »— ' 

oOTj-coTj-r^ 
lomioooON 

CNJ  VO  COCNJ  CO 

ooot^vom 

•-H  OOOO 

ooooo 


ooioo 

tOOr-H 

covoco 
lOCM  On 
vOOtT 

Tf  ■*  CO 

888 

odd 


0\V0  00 
ooo     00 

OOt^        '-H 

CO-"*  T-l 
O^O  CO 
COCN     CN 

OOO 

ooo 


Tfoiorgio 
coco^cviio 
ONCN  -^r^vo 
Onu-j  t^vo  On 

NO  vo  vor^  00 

COCM  r-H  O  On 

CM  eg  (N  eg  ^ 

ooooo 


'-H  CM  CO  CO  CO 
CO  CM  rj- ir>  Tj- 

rN.io  vot^to 

00  CM  OCMOO 
O  COVO  ONCM 
0\  00  t^^^ 

ooooo 
d>d>d>cid) 


OOt^Tj-Tt  T-H 

COOONOVOO 
tN.Oco  C0C7\ 

rv.  OLO  coco 

<0  '-tu^  OiO 
lO  lO  Tj-  Tj-  CO 

ooooo 


lO'-it^NCOON 

OCMOcot^ 
OOChi-HCM  T-H 

vor-Ha^ooc^ 
o  vD  '-<r>»co 

coCMCMi-n-H 

ooooo 
cf>d>cidic> 


vOOrooOOO 
cooocot^io 

00  "*  to  00-* 

i-ir^cocM  o 

O  coo  CAO 
'-•C^NCO  vovo 
t-H  OOOQ 
ooooo 


lOt^OO 

lO  cot^ 
CM  ^0\ 
CMiO  C7\ 
lorj-  CO 

ooo 
ooo 


VOOO    Th 

CMt^     ON 

1— I  o     to 
T-i  Ov     Cvj 

too    r^ 

coco     CM 

ooo 
ooo 

d>ci   d 


CM  to  1-1  CO  t^ 

CTncovOiovO 
OnvOO'OOO 
Tf  On —I  00  >-• 
VOiO^  VOOO 
rJ-coCM  '-lO 
CM  CM  CM  CM  CM 

ooooo 
(DcSddxD 


T-<iO00t^'«* 
CM  00  i^"^'-' 
CM  CM  VOO'-" 
OcoOCMt^ 
OCM  lOOO  '— ' 
OC7\00t^t^ 

ooooo 
c5d>cicic:> 


lOOOC^iCO^ 
t^OOCM  OOn 
lo  T-Hi^  On  t-^ 
lOt^  '-H  00  00 
to  ON  Tj- 00  CO 

vo^J^toT^  rj- 

ooooo 
ddddd 


CMCMCMTfvo 

VO'*  ON  00  ON 

On  cot^  T-'  CO 
QtO  1— I  O  O 
0\Tt  OVOCM 
CO  CO  CO  CM  CM 

OOOOO 


»— •  t^  tN.  Tf  in 

CM  coC^vCM  NO 

CO  CO  r^  NO  t^ 
CMtTNOCM  vo 

00  '-'t^  NO  NO 

i-iOOOt^^ 

r-H   r-H   ooo 

OOOOO 


00^^ 
Ot^CM 
to  O  CO 

to  NO  NO 

oO'-^in 

lOlOTf 

ooo 
ooo 

odd 


CMCM     Tj- 

o  o  t^ 
NOVO  CM 
"^  0\  o 
OLO  CM 
Tj-  CO  CO 
OOO 
OOO 


OO 


lOioCM  On  CO 

r^O'-<t^  NO 
ONOoor^co 

CMt^t^Ttt^ 

NO  to  to  NO  t^ 

lOTtcoCM  '-H 
CM  CM  CM  CM  CM 

OOOOO 

<:5d>d>c>c> 


CMt^t^OON 

00  OtoONC7\ 
OtoCM  ONCO 
to  t^  Tl- "Tj- C3N 
On— irM^O 

OOOnOOOO 
CMCM  ^  '-ht-i 

OOOOO 


c>>ot^Noo 

Ot^iOVOt-i 
CM'-hOvoOO 

r^oocvi  00 1^ 

Tj-OOcot^CM 
t^  NO  NO  to  to 

ooooo 


'-I  CO  NOON  Tj- 

CM  Tt  CO  NO  CM 

coOOOto  1— • 
ON  CO  00  NOVO 
t^coOOTfO 
'et  Tj- CO  CO  CO 

ooooo 
dic5ci<oci 


^ONOCMO 
CTNCMCOtoco 
CO'-'  CO  ON  ON 

t^  NO  to  r^  00 

NO  CJn  to  CO  CO 
CM  O  C7\CX)I%. 

ooooo 


Ocoto 
t~v.ONr^ 

to  O  CO 

Tt-CM  CTn 
tOOO'-H 
NO  to  to 

ooo 
ooo 


ooooo  ooo 


Orf  O 
NOrf  (50 
f^ON     00 

Tl-NO  Tf 
NO'-H  tX 
'^Tl-      CO 

ooo 
ooo 
d>c5   d 


OnOO-^On 
'-^co^Tj-OO 

cor^t^Tt-NO 

NO  to  to  NO  t^ 
NO  to  Tj-  COCM 
CM  CM  CM  CM  CM 

ooooo 


coot^coco 

NO^t  COi— I  NO 

COCM  '^  NO"* 

CO  to  t-H    r-H  to 

(?nt-h  Tj-r^  o 

'-"-iOOnOn 
CMCMCM  r-i  ,-H 

ooooo 


ONONOIOOO 
r-l  OCM  NO  CO 
t^r-   Tt  TtO 

CM  CONOCM  >-• 
Tt-OOCM  t^CM 

OOt^t^NONO 


NOtoCMOCO 
f^CMTl-Ot^ 

ON^HCOtO-*- 
r-H  tOOt^NO 

r^  CM  00  CO  ON 

lOtO"*""*  CO 


OOOOCOtMO 

NO  '-H  ■*  Onco 


ooooo 

d>CDd><0<D 


ooooo 


to  CX3  CO  >— I  »— I 

cO'-i  O  O\00 

ooooo 


CMCMVOcO'-H  r^ 

t^  COON  ^-H  00  to 

OOOCO'-H  NO  o 

O'^C^NCM  '-H  t^ 

cotoOOcoOO  CO 

r>vNO  tOlO  Tj-  Tj- 

88888  8 

dxz^dc^d  d 


"^CMt-inOI^ 

to  T}-  ON  00  00 
00  ON  l^  t^  CO 

t^oo^oo 

NO  NO  NO  NO  t^ 
t^  NO  to  ""^  CO 
CM  CM  CM  CM  CM 

ooooo 


^COO 

»-H  cor^ 

i-ctoCM 
to  NO  CM 
CA  ^  rf 
CMCM  '-H 
CM  CM  CM 

ooo 


CO  to 

00  NO 
On  CO 

r-H  to 

t^O 

oo 

CM  CM 

oo 
d>ci 


»-<CONONOt^ 
T*NO  ONOtO 
r-lOON-^tO 

CMCM  Tl-OOO 

n-oocMt^'-i 

ONOOCOt^t^ 


CM  to  NO  to  CO 

'-Ht^Ot^NO 
Q^-*  OtO 
0\  ^  NO  CO'— I 
NO  CVl  t^  CO  ON 
NONOtOtO  "* 


CO  to  CO  00  "-I 

NO  Tj- On  NO"* 
t^'-i  C7NCM  c^ 

i-itoooNo '-" 


^^^^^    ^^^^^    ^^^^o 

ooooo     OOOOO     OOOOO 


ooooo 
d>dd>cic> 


ooooo 
(Oc5d><5c> 


ooooo 


lOCMOOVONO  00 

CMtoOTt  NO  CM 

CO  NO  00  >-<  CO  Tj- 

CM  Tt-Nor^Tj-  t^ 

'-I  CO  NO  Oto  O 

001^  NONOtO  to 

88888  8 

ddddd d 


»-<  CO  C?\  Tl- t^ 

'-I  t>.  00  Tj- ON 
OOtOOt^ON 
Tft^t^CM  CO 

t^  NO  NO  t^  00 

cor^  NO  to  Tf 

CM  CM  CM  CM  CM 

ooooo 
ddddd 


OO^iOOQ 
NOcoOtO  VO 

CO-^OO'-H   t-H 

O  '-•NO  NO  ON 
OCM'^-t^O 
Tl-  COCM  '-"-" 
CM  CM  CM  CM  CM 

ooooo 
ddddd 


OCM  cot^'-i 

NOONOCOTf 

to  »-•  tot^  ■*■ 

to  to  t^  CM  O 

n- 00  CM  t^  CM 

OONONOOOO 
CM  -H^^^ 

ooooo 
ddddd 


lOtOl-lrt■r^ 

ot^'-ioor^ 

tOt^  '— I  CO  ■*" 

OCMl^co  '-" 
r^CM  t^  CO  On 
t^  t^  vo  NO  to 

ooooo 
ddddd 


CO  t^  to  t^  CM 

t^  CO-*  NO  00 
CM  NO  Tj- NO  CM 
'-I  CM  -^CTn  CO 
tot^  CM  On  C^ 

to  COCM  o  c^ 

'-I  —t  '-"-•O 

ooooo 
ddddd 


On  to  00 

ooQC^ 

NO  00  ON 

OCM  to 

ONCX)t^ 

Soo 
oo 
odd 


NOl-l  Ti- 

CO  to  to 

OtO  NO 

ON  CO  00 

NO  NO  to 
OOO 

oo  o 

dd  d 


Oi-iCMCOTf     lONOr^OOCTN     O'-<CMC0Tf 
CO  CO  CO  CO  CO    CO  CO  CO  co  co    ^  ^  ^  ^  '^ 


tONOt^00C3N     OtoOtoO     tootooto 

'*'*'^'*'«*   totoONotv.   ^N0000ONON 


354 


TABLES 


ON'-hON 

OcO(M'<:f -^ 

VO  rl-  CO  r-H   ,-4 

vOTj-O-rfvO 

mdlo  r^oo  o 

CM  LO  r>«.  to  1— t 

oooo-=i- 

Tf  vOOON<M 

OnOn  Oi-i  On 

r^  l-H  OOLO  00 

lOLO  LO  Tf  o 

Tf-rfLOOO  On 

t^  CO  '-^  LO  NO 

VOOn^ 

VO  CN)  to  lO  C<l 

i>.rsir^O'^ 

CMCMTf-voo 

NC^t^OslON 

COOC^ 

C>>VOCOCOCNl 

VO  OM^  MD  CO 

voLorrLocg 

OO  O  TM^t^ 

CM  o  OA  ON  r>s 

^ 

Tj-  i-HLO 

CO  CO  T-H  oo 

oor>.CM  ONLo 

On  ONTj-co^O 

1-hOOCM^ 

CM  On  NO  LOlO 

cT 

10-^00 

t^co  0\'— 1 1^ 

LOVO  OnCNII^ 

CM  OOLO  CM  On 

r^Lo  CO  l-H  c?\ 

00  NOLO  Tf  CO 

VO 

00  ^CNl 

tv.^T-|O00 

t^  ^OLO  LO  -rj- 

"^  CO  CO  CO  CM 

CM  (M  CM  CM  '-H 

o 

.Tj-cocvi 

OOOOO 

OOOOO 

OOOOO 

OOOOO 

a 

^ooo 

d>d>d>d>d> 

<5d>d><:5d> 

d  d  c3  d  o" 

<:5d>d>c5<5 

d>d>d>c^d> 

P4 

OOiOOO 

ot>>.rvi^oo 

00  o^l-Hr^^s. 

C?M-H  Tj-  CM  »-• 

On  ^  l-H  CM  O 

NO  CM  NO  CO  T-t 

00  Tf  ooooo 

Lo00Tj-r>.ON 

CM  On  ^  Cvi  o 

LO  l-H  LOCO  On 

Tl-  CO  OO  LO  lO 

■^  00  1 — 1 

T|-l^O\-HO 

Tj-OOLOLO  CO 

CM  OnOn^lo 

CM  VOONO  O 

CM  "^  1— '  CNJ  VO 

O  O  '— ' 

r^  »-<  '-H  c\j  c^ 

OOOCMloCV) 

Tf  VO  ON  Tf -rt- 

-^^  ONt^cot^ 

LO  NO  On  CM  "«}■ 

<:5 

OOCNIO 

OnOOOI^vO 

iOCOOOtJ-O 

CO  CM  NOLO  r^ 

<M  ON  c:tn  T-H  Tt 

On  LO  CNj  ,-H  o 

w 

t^t^CM 

Ot^CN^O 

0\OCM  VOi-H 

MD  CM  OOLO  CM 

Of^LO-^CM 

O  On  X  t>»NO 

OO'-HCO 

00  Tl- (MOON 

t^r^  vololo 

■rt-^  coco  CO 

CO  Cvi  CM  CM  CM 

Tj-  cocg 

r-i^^^O 

OOOOO 

OOOOO 

OOOOO 

OOOOO 

o 

T-^dod 

d>d>d><:5d> 

d>d>d>d>d> 

c5d>d>d>d> 

<:Dd>d>d>d> 

cid>d>d>(6 

Q 

vovoio 

TfONt^lOt^ 

<N00  ONLOCNI 

1— it^OO  Or}- 

■^r^inoNco 

cor^NO^»-M 

w 

^ 

lOCOMD 

^cOTj- vo-^ 

00 '-H  l-H  coco 

CC  CO  LO  ON  CO 

i-HioNOrJ-  O 

O  CO  Tf-OOVO 

t^  COCO 

-•OO-^CATJ- 

CO  00  VO  LOO 

CO  LO  i-^  vo  r^ 

vooLoc<ir>. 

On^OnO^ 

;?2 

ONt^Tj- 

OM^  OOt^ 

t>»cox^\ou^ 

t^Tf\OJ>sC\l 

l-H    l-H    l-H    coo 

r>.  O"^ooco 

CO  CM  ^Cg  l-H 

w 

ONt^r^ 

COCVI  VOCM  OO 

l-H  OTj-CM  Ti- 

00  VDlovOOn 

^  Ot^LOrJ- 

^i- 

00  00  CO 

C^OOtJ-vOCN 

-H  CM  rf  00  <N 

ooTfor^^ 

l-H  CTn  t^lO  CO 

CM  -H  ON00t>x 

w 

00-^  CO 

OO^i-CvlOON 

001>.  VOLOLO 

Tt- Tj-  Tj-  CO  CO 

CO  CM  CM  CM  CM 

CM  CM  '-'  ^  '-I 

^    _ 

.Tt  tOC^l 

^^^^o 

OOOOO 

ooooo 

OOOOO 

OOOOO 

"-^diQd) 

o  c5  o"  c5  d 

d>d>d>d>d> 

c>c5c5d)d> 

d>d>d>cSd> 

d)d>d>d>d> 

c^ 

o5)0 

^0-Hf0  0\ 

Tj- rj- t>^  CO  t^ 

OOCM  coCM 

lOl-H  l-H  VOCO 

vooOTj-ooeo 

-hONVO00O\ 

onOi— ir^ON 

l-H  OLOCONO 

t-^i-HOOOCO 
■■-H  OCOOnMD 

ON  COLO  OnCJN 

H 

^^s 

t^  r-l  ONl^CNI 

l-H  OOOCOOO 

l-H  t^  00  CM  On 

< 

CNJ  VO  OCM  On 

ON  ^  LO  ^  VO 

Tj- CM  On  On  CO 

COCOOnOOO 

l-H  NO  CO  l-H  tN, 

{fX        r. 

^ 

1—*  CO  '^ 

vOt>^  vQiO-^ 

O\00  ^  On  T-H 

lOCN  '— 1  COLO 

OLoCslOOO 

l-H       O 

OOlo 

'f  o  ^oo"^ 

CO-^OO"^ 

ONLOCM  OOMD 

CO^  ONt^LO 

Tf-CM^OOO 

P-. 

-* 

OnCVJ  CO 

00  LO  C^l  o  o\ 

oor^  vo  voLo 

rj-rj-^coco 

CO  CO  c^j  CM  r^i 

C^J  CVl  CM  CM  »-" 

^TffOCM 

^^^^o 

OOOOO 

OOOOO 

OOOOO 

OOOOO 

r-^d>d>d> 

ddddd 

d><:5d>d>ci 

d>d><zSd)d> 

<d>d>d><5d> 

d>c5<z^d)d) 

> 

w 

K  < 

OnoO'* 

1>»^ONLOt-< 

tv.t^iot^ro 

t^  ^Ti  tn -rt- t^i 

OOONt^Oro 

'^O'-'inco 

>_J 

u 

coCNTt^O 

coONONtor^ 

O  00  '-H  CO  CO 

OLOOOOOO 

O'^'Tj-NOCO 

pq 

•^    J 

o  com 

^OOTif^'O 

^  ,—1  CO  l-H  o 

LO  '^  CO  vo  O 

l-H  \OCMOOCM 

rJ-LOCMCMLO 

K    -] 

^ 

oovc^r^-^ 

Ti-ONOO^r^ 

(M  00  Tf  r-H  T:f 

NO  CO  CO  l-H  t^ 

t^OLOOTf- 

< 

TfONCNJ 

■^  \0  >-0  "'^  "^ 

CNJ  0"^OLO 

LOMD  OOCN  VO 
OOt^  VOVOLO 

OO^OCO  On 

cOOOnO  CM 

NO  CNl  00  NO  ^ 

H 

^^ 

^ 

VOCM  OOO  VO 

-Ht^Tj-Ot^ 

lOCOOCM^ 

LOTi-CM-HO 

On  CO  CO 

OOLOCM  '-'  0\ 

LO  Tj- Tt  rt  CO 

CO  CO  CO  CM  CM 

CM  CM  CM  CM  CM 

CO 

-^COCM 

1—i   T— I  ,—(   l-H  O 

OOOOO 

OOOOO 

OOOOO 

OOOOO 

r-^d>d>d> 

d>d)d>c5d> 

d><zic^d>d> 

d>d><:5cid> 

<z>c5>d>d>d> 

d>c5d>d)d> 

'^^O^JO 

t>sOlOO^  vo 

'-iLOONTh^ 

LOTtOLOCO 

OOLOCOOOQ 

LOOOLOt^OO 

r-HOOONOCM 

!z; 

00  coo 

lO  to  CO  CO  00 

t^  t^  CO  On  •^ 

00CN|CM(MVO 
tv.OOrfcO'^ 

^ 

oot-^ 

tM^vo^co 

Ot^CN  On^ 

VOOCM  00  CO 

LO^  t^  COt^ 

'-I  co(M 

lO  ONOLO  CO 

cor^  vocM  eg 

VO-^LOO-H 

l-H  r^  xt- l-H -^ 

CM  CO  NO  ON  »-H 

< 

vOloo 

COIOIO'^''^ 

CM  O-Tj-OLO 

t^vo  ONi^co 

CM  OOt^  OOO 

Tt  ONLOCM»-t 

^ 

C<lcoON 

OO^OCMCO 

r^ooo'^oo 

CO  ON  LO  CM  ON 

t^TTCMOON 

t^uT^coCM 

0\CN>  CO 

OOiOcO'— 1  ON 

oor^t^  vOLo 

LO  "^  -rf  T}-  CO 

CO  CO  CO  CO  CM 

CM  CM  CM  CM  CM 

« 

CO 

.^COCNl 

^^^^o 

OOOOO 

OOOOO 

OOOOO 

OOOOO 

o 

Q 

y-*d><z^d> 

d)d>d>d)d> 

d>d)d>d>d> 

d><:5d>(od> 

<D<5d>d>d> 

d>d>d>d)0 

ipcOON 

Cv)  CO  t^  LO  m 

CMONr-^CMt^ 

t^OvOcoCM 

cOt-<OOco 

t^  NO  NO  00  to; 

CMT:^lO 

cO00rfO\ON 

t^CMt^LOLO 

tv.TtM--HMD 

CJN  l-H  t^  CO  CO 

Ph 

^ 

00  (N  O 

ooot^r^o 

OnVOOOCM  -^ 

On  t~>«  l-H  o  00 

'-HONNOTJ-OO 

On  1-Ht^r^ON 

l-H  COCN) 

ONt^  ONLO  t:}- 

cooo^cocg 

LO  0\  COCOf^ 

cOTfLOt^OG 

^ 

rqcoON 

CMuT^^rf 

CM  O^OLO 

t^LO  On  ^t^ 

1-HOOVOt^ON 

coco-^-Hc:rN 

o 

COT;J-a\ 

Onlo  1— 1  coo\ 

00  On^loOn 

oot^r^  voLo 

■"^  O  ^coO 

OOLO  CO '-H  ON 

OONOLOTj-CM 

^ 

C<1 

0\(M  CO 

OOLO  CO^  o\ 

LOLO  T^  Tj-  Tj- 

CO  CO  CO  CO  Og 

CM  (M  CM  CM  CM 

Tf  coCN 

OOOOO 

ooooo 

OOOOO 

OOOOO, 

/"N 

g 

in 

'-^dcJd 

cDd>d>d>d> 

d>d>d>d><z^ 

d>d>d>d>d> 

d><5d>d>c^ 

d><od>d><^\ 

00 

en 

O 

^CSICO"^ 

lOVOtxOOON 

OT-iCMrOrJ" 

lOVOI^OOOv 

O^CMCO'^t 

lovorvooON 

CO 

T— t  t— <   T-<  T— 1  T-< 

»— 1 1— 1 »— 1 1— ( »— t 

CM  CM  CM  CM  CM 

CMCMCMCvlCM 

609 

p^ 

COMPOUND    INTEREST;    OTHER     COMPUTATIONS 


355 


OnCVI  roONTt 
OOCN  (VI  (Nl  00 

CVl  ^  — <OC7N 

ooooo 

OC>OOCJ 


ooooTf^r>^ 
coTt-r^ooco 
t^  OMOu^  On 

On  coco  ro  00 

oooot^r^  vo 
ooooo 
ooooo 

OOCJOO 


TtvOCMCM  vo 

inoo-^  ^  o 

'-H  OOcorO  VO 
VOLOOO  ro  O 
Tto^<^0 

,VO  ^  LO  VO  lO 

ooooo 

OOOOO 

ooooo 


Omoovo  vo 
to  00  VOvQiO 

o  -^t^t^  fO 

O'-HTt-ON^ 


ooooo 


oworvi  vofo 

CVJ  Ost^vO'-i 
Tf  \Ovo  OfO 
tT  cot^  ONfO 
TfiOOOcoO 

roCM  ^  T-._ 

ooooo 
ooooo 
ooooo 


r^rf^vooo  vo 
voiooofom  fo 
oocvj vooot^    r>^ 

VOt^C^  T-H  CO     t^ 

80Q00 
ooo  ~ 
oooo 
ooooo 


Tf  CM  rvj  Tjr  to 

»-iCMOOiO 

»Oco00O\»O 

O^CVjTtt^ 

lO  Tf  CO  eg  >-• 


'-"<oo\<^rM 

t^Tft-xCVl\0 
'-1  ''I- ON  -"^  Tj- 

IN»  corOOO  ^ 
OTfCOCvit^ 
'-1  OOnOnOO 


OOOOO  ooooo 


ooooo 
ocJooo 


voOn'— irom  too  T-ir^\n  rfvoootoio  »— i  rvj  vo  »- '  fO 

^CN)r>vco(vj  t>v(Nirvj'rfvo  r>.00'-i^^  vo^i— "-'O 

OOCM-^roNO  '-'OO'^OOOn  vo  nO  00  ON  On  ^  On  ro  rv  Q 

r^CMONON^  vocMt-H^co  t^NOCMOOON  cgcMOCMOv 

CVJOOcoOnvO  CVJONVOcoO  t-^VOOO'-'^  fOOOOvOTt 

OOt^t^NONO  "^  \J-)  xn  xn  in  tj- ro  CN  CM  »- 1 

ooooo  ooooo  ooooo 

ooooo  ooooo  ooooo 


ooooo  ooooo  ooooo  ooooo  ooooo  o 


TfvoOfO'-H 
tOTt  CMioON 
'-'fororr^ 

ONTj-NO'<^00 
COTflOt^  0\ 
VOlOTf  COCVJ 


ooooo 

OOOCJC) 


lOOOCMONt^ 
''^•t^O  vo  NO 
OiO  -^  .— iirj 

i>.O00OiO 
CM  VOONTf  00 
CM  -hOOOn 

ooooo 
c>  o  o  o  o* 


lOOOOOiOi-t 

»-H  lO  VC  fO  t^ 
CO  '-I  00  CV)  o 
'^NOOOOOO 
ro  CO  Tj-  On  «-0 
On  oocot^r^ 


OOOOO 


CM  '-H"«^OOCM 

O  r^  foio  CM 

CM  rft^OOt^ 

OTf  ooooo 

CM  OOiO— '00 
t^  vo  NO  ^«J^ 

8  oooo 
oooo 
dddoo 


iOTj-vorx'-< 

r-HloCM  Tfr^ 

CM  t^  -"^  OiO 

O  OO^^fO^ 

NO  fO"^tN.  1— ( 
uo  Tl-  fOCM  (M 


ooooo 


Tj-O\rj-V0  a\  o\ 

O  VO  CO  '-'  On  CO 

'-C  O  CO  fOt^  00 

CM  l^  ONt^  ON  lO 

r>Nfoooovo  lo 

ooooo  o 

ooooo  o 

c5<ocSc^(zi  d 


OiOO\t>.t^ 
.-.  coioirjt^ 
O^OOO  cOTj- 
coiOTj-  O  '— < 

00  00  On  ^  CO 

t-^  NO  VOIO  -^ 

ooooo 
<:5<od)d><:D 


CM  00  tN.  CM  fO 

rooOioO\00 
r^  voon  •-I  o 

t^OO  fO  CO  NO 
lOOOCM  no  O 
CO  CM  CM  '-"-I 

ooooo 


ONOOOONTf 
Tj-fOCM  OOiO 
fOt^  O  OntJ- 

CM  '-I  n-ooNo 

to  0>J-)  C  NO 
OOONONOO 
'—'-^  OQ  O 

ooooo 


VOIO  OnioCM 

TfOOONO'-H 
CM  CM  r-OI^ 
NO  00  CM  00  vo 
CM  OO^O  T-i  00 

oor^i^r^No 

ooooo 
ooooo 


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CM  CM  00^  O 
0'-"^OiO 
lo  fO  O  On  Ti- 
lO  CM  CM  CO  tN. 
VOiO  Tl-  coCM 


qoooniooo  q 

Q  O  Ot^  CO  O 

OvTj-csr^t^  00 

CM  '-HtN.ooO  O 

CMOOtI-CM  On  00 


CM   '-H   ,-H   ^   O  O 

O  QOQO  O 

ooooo  o 

cSd>'^<:5<zi   c5cid>ci<5  d 


—•CM  '-'CM  ON 

t^  r>.  •'I- 1^  lo 

CO  CO  "^  to  t^ 

On  001^  NO  to 

ooooo 


lovotOTj-m 

COt-iCM  "-H  t^ 
00 '^  CO  CM  t^ 

ON  00  r-HOOOO 
OnOvJ  NO  On  CO 
■«:1- Tf  CO  CM  (M 

ooooo 


00  CM  00  On  00 
CM  CM  CM  CO  NO 

t^  00  00  to  t^ 

CM  On  ON  CM  t^ 
00  CM  t^  CO  00 

;ij;i^22o 

ooooo 


CO  00  NO  NO  t^ 
TfO-^o^NO 

CO  "-I  owe-* 

ti-5  »0  NO  O  NO 

rfO  NO  CO  0\ 
On  On  00  00  t^ 

OOOOO 

ooooo 


^-HCOCMvOtO 
t^  CM  NO  CM  On 
CO  CO  00  00  o 
CO^OOOONO 

NocM  o^-i  T^ 

tN»  NO  tOTf  CO 

~    "    "        o 

o 
<ziocD<od> 


o  oncm  '— '  vo   rx 

'-'OONOOOTf  CM 

C?N  rf  NO  t^to  On 

NOOOOOtOOO  to 

00  CO  On  NO  CO  •-I 

CMCM  '-"-H  ^  ^ 

88888  8 

ddddd   d 


vOfOCMCMVO 

CM  On  NO'-"  ON 
On  00  NO  ^'—' 

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^  On  On  00  t^ 

(M— .^r-^-H 

ooooo 


On  On  CM '^  to 

CM  t^  NO  CO  00 
C3n  CO  »-'  On  CO 
coo  '—I  to  Tj- 
tOOO  '-H  "^00 
NO  to  to  "^  CO 


OO^t^T-ito 

CO  rj-  NO  '-H  00 
CMCM  '-'OOCA 
NO'-H  OnCTnCM 
CMt^'-iNOCM 
COCMCM  '-H  »-< 


ooooo 


ooooo 


,—1  lO  to  l^  i-H 

to  CM  Ot^  CO 

OONO  NOt^  r-l 

r^  CO  On  to  CM 

O  O  On  On  ON 

ooooo 


or^  VD  "-H  CO 
to  O  On  00  vo 
to  ON  CM  to  VO 
VO"^  corf  CO 

00  CO  '-H  r-H  CO 

oo^^  NOtoT^ 

88888 


NOioovor^   r^ 

ON  tN.  to  tOt^      vo 

tN» ,— 1  vototo    vo 

NO  '-1  Tl-IOCM      Tf 

vo  '-H  VOCM  On     vo 
COCOCMCM  i-H     ^ 

88888  8 


ooooo  ooooo  o 


CM  CO  CO  CO  to 
^  to  NO  to  t>» 

Tj".*- CM  CM  00 
00  to  On  On  -^ 

OOOOOOON'-" 
— 'OONGOOO 
CM  CM  •-'  ^  '-' 

ooooo 


to  CM  CO 

■^  COLT) 

NO'—  ON 

to  T- '  o 

CO  NO  ON 
t^  NO  to 

ooo 


Tj-NO 

NO  to 

r^CM 

CM  NO 

to  Tf 

So 
d>cS 


T-HQtO'-i  O 

to  or^i^  O 

'-'  Cvj  '-'  00  '-H 
coi^  Tf  CO  vo 
O'^ONTfC^ 
Tf  coCM  CM  '-' 

ooooo 


CO  CO  00 
On  On  to 

NOTf  CO 

or^  NO 

to  O  vo 

ooo 


00  CO 
tot^ 

t^  0\ 

C^lOO 
O  0\ 
'-'  o 

o  o 


CM  CO  CM 

ONtO  O 

OOnO 
Tj-t^CM 

to  ONt^ 

ONt^  NO 

ooo 
ooo 


OOO 

(M   -H 

^CM 

ON'rf 
NO  00 

to  Tf 

88 


QCMOtO'-H  CJO 

NOTf  CM  CM  Tf  1— 1 

tOCOTf  ^  ,— I  Tf 

CO  Tf  Tf  CM  NO  to 
'-HtoOvOCM 
Tf  CO  CO  CM  CM 

8 oooo  o 

oooo  o 

d>d>d>cid>  d 


ON 


O'-'CMCOTt     iONOt^00C?N     O^CMCOT^     to  vO  t^  00  C?\     O  to  O  to  O     to  O  to  O  to 
COCOCOCOCO      COCOCOCOCO      T^TfTfTfTf      TfTfTfT^fTf      lOtONONOt^.      t^  00  00  On  ON 


356 

§383. 


TABLES 


Table  VI 
Reciprocals  and  Square  Roots 


Square 

Square 

Ratio 

Reciprocal 

Root 

Ratio 

Reciprocal 

Root 

(1  +  0 

(Discount 

(Quarterly 

(1+0 

(Discount 

(Quarterly 

Multiplier) 

Ratio) 

Multiplier) 

Ratio) 

1.005 

.99502488 

1.00249688 

1.02 

.98039216 

1.00995049 

L00525 

.99477742 

1.00262156 

1.02025 

.98015192 

1.01007425 

1.0055 

.99453008 

1.00274623 

1.0205 

.97991181 

1.01019800 

1.00575 

.99428287 

1.00287088 

1.02075 

.97967181 

1.01032173 

1.006 

.99403579 

1.00299551 

1.021 

.97943193 

1.01044545 

1.00625 

.99378882 

1.00312013 

1.02125 

.97919217 

1.01056915 

1.0065 

.99354198 

1.00324474 

1.0215 

.97895252 

1.01069283 

1.00675 

.99329526 

1.00336932 

1.02175 

.97871299 

1.01081650 

1.007 

.99304866 

1.00349390 

1.022 

.97847358 

1.01094016 

1.00725 

.99280218 

1.00361845 

1.02225 

.97823429 

1.01106380 

1.0075 

.99255583 

1.00374299 

1.0225 

.97799511 

1.01118742 

1.00775 

.99230960 

1.00386752 

1.02275 

.97775605 

1.01131103 

1.008 

.99206349 

1.00399203 

1.023 

.97751711 

1.01143462 

1.00825 

.99181751 

1.00411653 

1.02325 

.97727828 

1.01155820 

1.0085 

.99157164 

1.00424101 

1.0235 

.97703957 

1.01168177 

1.00875 

.99132590 

1.00436547 

1.02375 

.97680098 

1.01180532 

1.009 

.99108028 

1.00448992 

1.024 

.97656250 

1.01192885 

1.00925 

.99083478 

1.00461435 

1.02425 

.97632414 

1.01205237 

1.0095 

.99058940 

1.00473877 

1.0245 

.97608590 

1.01217588 

1.00975 

.99034414 

1.00486317 

1.02475 

.97584777 

1.01229936 

1.01 

.99009901 

1.00498756 

1.025 

.97560976 

1.01242284 

1.01025 

.98985400 

1.00511193 

1.02525 

.97537186 

1.01254630 

1.0105 

.98960910 

1.00523629 

1.0255 

.97513408 

1.01266974 

1.01075 

.98936433 

1.00536063 

1.02575 

.97489642 

1.01279317 

1.011 

.98911968 

1.00548496 

1.026 

.97465887 

1.01291657 

1.01125 

.98887515 

1.00560927 

1.02625 

.97442144 

1.01303998 

1.0115 

.98863075 

1.00573356 

1.0265 

.97418412 

1.01316336 

1.01175 

.98838646 

1.00585784 

1.02675 

.97394692 

1.01328673 

1.012 

.98814229 

1.00598211 

1.027 

.97370983 

1.01341008 

1.01225 

.98789825 

1.00610636 

1.02725 

.97347286 

1.01353342 

1.0125 

.98765432 

1.00623059 

1.0275 

.97323601 

1.01365675 

1.01275 

.98741052 

1.00635481 

1.02775 

.97299927 

1.01378006 

1.013 

.98716683 

1.00647901 

1.028 

.97276265 

1.01390335 

1.01325 

.98692327 

1.00660320 

1.02825 

.97252614 

1.01402663 

1.0135 

.98667982 

1.00672737 

1.0285 

.97228974 

1.01414989 

COMPOUND   INTEREST;    OTHER    COMPUTATIONS 


357 


Reciprocals  and  Square  Roots — (Conclvuled) 


Square 

Square 

Ratio 

Reciprocal 

Root 

Ratio 

Reciprocal 

Root 

(1+0 

(Discount 

(Quarterly 

(1+0 

(Discount 

(Quarterly 

Multiplier) 

Ratio) 

Multiplier) 

Ratio) 

1.01375 

.98643650 

1.00685153 

1.02875 

.97205346 

1.01427314 

1.014 

.98619329 

1.00697567 

1.029 

.97181730 

1.01439637 

1.01425 

.98595021 

1.00709980 

1.02925 

.97158125 

1.01451959 

1.0145 

.98570725 

1.00722391 

1.0295 

.97134531 

1.01464279 

1.01475 

.98546440 

1.00734800 

1.02975 

.97110949 

1.01476598 

1.015 

.98522167 

1.00747208 

1.03 

.97087379 

1.01488916 

1.01525 

.98497907 

1.00759615 

1.0325 

.96852300 

1.01612007 

1.0155 

.98473658 

1.00772020 

1.035 

.96618357 

1.01734950 

1.01575 

.98449422 

1.00784423 

1.0375 

.96385542 

1.01857744 

1.016 

.98425197 

1.00796825 

1.04 

.96153846 

1.01980390 

1.01625 

.98400984 

1.00809226 

1.0425 

.95923261 

1.02102889 

1.0165 

.98376783 

1.00821625 

1.045 

.95693780 

1.02225242 

1.01675 

.98352594 

1.00834022 

1.0475 

.95465394 

1.02347447 

1.017 

.98328417 

1.00846418 

1.05 

.95238095 

1.02469508 

1.01725 

.98304252 

1.00858812 

1.0525 

.95011876 

1.02591423 

1.0175 

.98280098 

1.00871205 

1.055 

.94786730 

1.02713193 

1.01775 

.98255957 

1.00883596 

1.0575 

.94562648 

1.02834819 

1.018 

.98231827 

1.00895986 

1.06 

.94339623 

1.02956301 

1.01825 

.98207709 

1.00908374 

1.0625 

.94117647 

1.03077641 

1.0185 

.98183603 

1.00920761 

1.065 

.93896714 

1.03198837 

1.01875 

.98159509 

1.00933146 

1.0675 

.93676815 

1.03319892 

1.019 

.98135427 

1.00945530 

1.07 

.93457944 

1.03440804 

1.01925 

.98111356 

1.00957912 

1.08 

.92592593 

1.03923048 

1.0195 

.98087298 

1.00970293 

1.09 

.91743119 

1.04403065 

1.01975 

.98063251 

1.00982672 

1.10 

.90909091 

1.04880885 

INDEX 

(References  are  to  sections  unless  otherwise  noted.) 


Account, 

Amortization,  §§  198,  205,  208-212,  214. 

Insurance,  §  179. 

Principal,  §§  164,  169,  179,  186-188,  202. 

Taxes,  §  179. 
Accounts, 

Amortization,  installation,  §§  334-336. 

Bond  and  mortgage  loans,  §§  162-184. 

Bonds,  §§  197-214. 

Collateral,  §§  185-188. 

Discounts,  §§215-221. 

Interest,  §§  161,  164,  165,  171,  172,  174-176,  179-181,  183,  186-196, 
203,  204,  220. 
Accumulation, 

Dual  rate  for,  §§  326,  327. 

Schedule  of,  §  122. 
Amortization,  §§249,  348. 

Account,  §§  198,  205,  208-212,  214. 

Accounts,  installation,  §§  334-336. 

Definition  of,  §  70. 

Development  of  series  of,  §§  72,  249. 

Interest-difference,  §271. 

Of  premiums,  §§  328-330. 

Relation  to  present  worth,  §  71. 

Relation  to  sinking  fund,  §  90. 

Schedules  of,  §§  121,  122,  126,  130,  134,  139-141,  281. 
Amount,  §§25,  26,  28,  30,  31. 

Of  ordinary  annuity,  §§  53-61. 

Of  prepaid  annuity,  §  75. 

Proof  of  by  reciprocal,  §  227. 
Amounts  of  $1,  table  of,  §  378;  comment,  §  367. 
Amounts  of  annuity  of  $1,  table  of,  §  380;  comment,  §  370. 

359 


360 


INDEX 


Annualization,  §§  301-310. 
Annuities,  Chs.  IV-VII. 

Amount  of  ordinary,  §§  53-61. 

Finding,  §§60,  61. 
Analysis  of  payments,  §  68. 
As  sinking  funds,  §§  87-90. 
Deferred,  §§  11,  78. 
Deferred  payment,  §  86. 
Definition  of,  §  54. 
Due,  §§  75,  76. 

Immediate  or  ordinary,  §§  74-76. 
Instalments  of,  §  69. 

Present  worth  of  ordinary,  §§  62-73,  304. 
Problems,  §§  232,  233. 
Rents  of,  §§  83-86,  373. 

Problems,  §§  234,  235. 
Tables, 

How  formed,  §§  56-58,  257. 

Of  amount,  §380;  comment,  §370. 

Of  present  worth,  §381;  comment,  §371. 

To  four  periods,  §  63. 
Varying,  §82. 

B 

Base  in  logarithms,  §  391. 

Bond  problems,  Chs.  XXI-XXVII. 

Broken  initial  and  short  terminal  bonds,  §§255,  256. 

Cash  and  income  rates,  §  243. 

Compound  discount,  §  261. 

Compound  interest,  §  262. 

Discounting,  §§  270,  271. 

Income  rate,  accurate,  §§  263-269. 

Initial  book  values,  §  247. 

Nominal  and  effective  rates,  §§  236-240. 

Premium  and  discount,  §  243. 

Present  worth,  finding,  §§258-261. 

Redemption  of  bonds,  §§  287-294. 

Semi-annual  basis,  §  300. 

Serial  bonds,  §§  274-286. 

Successive  amortizations,  §  249. 

Successive  method,  §§  244,  318. 

Tabular  methods,  §  283. 

Varying  rates,  §§  298,  299. 

Varying  time  basis,  §  312. 


INDEX 

Bonds,  Chs.  IX-XI,  XVII,  XXI,  XXII,  XXV-XXVIII. 

Accounts,  §§  197-214. 

As  trust  fund  investments,  §§  148-154. 

Cullen  decision,  §§  152-154. 
Broken  initial,  problems,  §§255,  256. 
Cost  and  par  of,  §  104. 
Cost  of,  relation  to  net  income,  §  99. 
•   Discounts  on,  §§  207,  214,  258. 
Earning  capacity  of,  §  103. 
Elements  of,  §  110. 
How  designated,  §  98. 
Interest.     (See  "Interest.") 
Investments  in,  §  119. 
Investment  value  of,  §§  106,  108. 
Irredeemable,  §§  146,  213. 
Last  half-year  of,  §  144. 
Ledger  for,  §§  200,  202. 
Loans  on,  §§  162-181. 
Premiums.     (See  "Premiums.") 
Present  worth  of,  §§  101,  103,  106. 
Problems.    (See  "Bond  Problems.") 
Provisions  of,  §  96. 
Purchase  of,  §§  102,  128,  129. 

Adjusting  errors,  §  129. 
Rates, 

Annual  and  successive,  §§  295-318. 

Income,  §§  134,  136-140,  263-269,  287. 

Interest,  §§  97,  100,  109. 

Varying,  §  150. 
Redemption  of,  §§  146,  147. 

Problems,  §§  287-294. 
Repayment  and  reinvestment,  §§319-336. 
Replacement,  §§  148,  322,  323. 
Residues,  eliminating,  §§  138-140,  280. 
Schedules, 

Accumulation,  §  122. 

Amortization,  §§  121,  122,  126,  130,  134,  139-141,  281. 

Checks  on  accuracy,  §  124. 
Serial,  §  145. 

Problems,  §§  274-286. 
Short  terminal,  §§  141,  142. 

Problems,  §§  255,  256. 
Tables,  §  155;  comment,  §  156. 

Of  differences,  §  276. 


361 


362  INDEX 

Bonds  (Continued) 

Tables  (Continued) 

Use  in  compound  interest  problems,  §§  257-262. 

Use  in  determining  accurate  income  rate,  §§  263-269. 
Valuation  of,  Chs.  X,  XI. 

First  method,  §§111-114. 

Interpolation  method,  §§  131,  132. 

Multiplication  method,  §§  133,  134. 

Periodic,  §§  130-133. 

Problems,  §§  243-318. 

Rule  for,  §§305,  316. 

Schedules,  §§114,  122,  126,  127,  130. 

Second  method,  §§  115-120. 
Values  of,  §  127. 

Book,  §§  123,  125. 

Found  by  discounting,  §§  143,  144. 

Initial  book,  §  247. 

Intermediate,  §  105. 

Market,  §  107. 

Various,  §§  104-108. 

C 

Capital, 

Account,  §§  1,  3. 

Cash,  §  2. 

Definition  of,  §  1. 

Potential,  §  2. 

Sources  of,  §  3. 

Use  of,  §  2. 

Working,  §  2. 
Card  records  for  mortgages,  §  178. 
Cash,  §  2. 

Characteristic,  §41. 
Collateral,  loans  on,  §§  185-188. 
CompoundMiscount,  §§  33-35,  261. 
Compound  interest,  §§  13,  17-19. 

Amount  of,  §  28. 

Problems,  §§  225,  226. 

Interpolated  values,  §  376. 
Use  of  tables,  §§  262,  368. 

Rules  and  formulas,  §§35,  157-159. 

Tables,  §§  359,  360,  378. 

Use  of  logarithms  in  computing,  §  49. 
Contracted  division,  §  248. 
Contracted  multiplication,  §  228,  270. 


INDEX  363 


Conversions  of  rates,  §  92. 
Coupons,  §§  113,  296. 
CuUen  decision,  §§  152-154. 


Day  as  time  unit,  §§  20,  24. 

Day  basis,  360  and  365  methods,  §§  23,  24. 

Days,  odd,  how  reckoned,  §  23. 

Deferred  annuities,  §§  11,  78. 

Differences,  table  of,  §  276. 

Differencing, 

Discovery  of  errors  by,  §§250-254. 

Present  worth  by,  §  259. 
Discount,  Chs.  XVIII,  XIX. 

Compound,  §§  33-35,  261. 

Formulas,  §  35. 

On  bonds,  §§  207,  214,  258. 

Single,  §§33,216-218. 
Discounting, 

Contracted  methods  for,  §§  270-273. 

To  find  bond  values,  §§  143,  144. 
Dividends,  §  7. 
Division, 

By  logarithms,  §45. 

Contracted,  §  248. 
Dual  rates,  §§  326-333. 

E 

Effective  rates,  §§  91-95. 

Problems,  §§236-240. 
Errors,  discovery  of,  by  differencing,  §§  250-254. 
Evaluation, 

Method  by  logarithms,  §  73. 

Of  a  series  of  payments,  §  53. 
Exponents,  §§  27,  38,  39. 

Fractional,  §  48. 

F 
Factors,  logarithmic,  table  of,  §358;  comment,  §§342-357. 
Forms, 

Bond  accounts,  §§  197-214. 

Discount  accounts,  §§  215-221. 

Interest  accounts,  §§  176,  179,  188,  192,  193,  195,  220. 

Loans  on  collateral  accounts,  §§  185-188. 

Mortgage  accounts,  §§  176-183. 


3^4 


INDEX 


Formulas  for  interest  calculations,  §§  157-159. 

Fractional  exponents,  §  48. 

Frequency,  definition  of,  in  interest  computations,  §  10. 

G 

General  ledger,  §§  160,  180,  182,  195,  200. 
Gray's  tables  of  logarithms,  §  341. 

H 
Half-year,  legal  definition  of,  §  22. 


Immediate  annuities,  §§  74-76. 
Income  rate  on  bonds,  §§  134-140,  263-269,  287,  297. 
Dual,  §  326. 
Relation  to  cost,  §93. 
Rule  for  determining,  §  293. 
Use  of  tables,  §§  257-259. 
Increase,  ratio  of,  §  17. 
Initial  book  values  of  bonds,  §  247. 
Insurance  account,  §  179. 
Interest,  Chs.  II,  XVI,  XVIII,  XIX,  XXI. 

Accounts,  §§  161,  164,  165,  171,  172,  174-176,  179-181,  183,  186-196, 

203,  204,  220. 
Calculations,  formulas  for,  §§35,  157-159. 
Compound.    (See  "Compound  Interest.") 
Computations,  §§  15,  19. 
Constant  compounding,  §  238. 
Contract,  essentials  of,  §  10. 
Definition  of,  §§  6,  9. 
Equivalent    rates    of     (annual,    semi-annual    and    quarterly), 

§§241,  242. 
Laws  of,  §  8. 

One  per  cent  method,  §  223. 
Periods,  §§  20-24. 
Punctual,  §§  14,  17. 
Rates, 

Coupon  (cash),  §§  100,  109. 
Diminishing,  §§  323,  324. 
Dual,  §§  326-333. 
Effective,  §§  91-95. 
Nominal,  §§91,  97. 
Usual,  on  bonds,  §97. 
Varying,  §  325. 


INDEX 

Interest  (Continued) 

Ratios,  table  of,  §359. 

Receipts  and  notices,  §  181. 

Register,  §§  180,  183,  192,  193. 

Simple  (single),  §§  13,  16,  19. 
Problems,  §§222-224. 
Interest-difference,  present  worths  of,  §  271. 
Intermediate  dates,  bonds  purchased  at,  §  128. 
Interpolated  method  of  valuation  of  bonds,  §§  131,  132. 
Interpolation  in  interest  problems,  §§  339,  376. 
Inter-rates,  §§  275,  375,  376. 
Investment,  §§4,  119,  150. 

Absolute,  §4. 
Investment  value  of  bonds,  §§  106,  108. 
Investments,  trust  fund,  bonds  as,  §§  148-154. 
Irredeemable  bonds,  §§  146,  213. 


Ledger, 

Bond,  §§  200,  202. 

Books  auxiliary  to,  §  173. 

Forms,  §§  182,  195,  200. 

General,  §§  160,  180,  182,  195,  200. 

Loose-leaf,  §  178. 

Modern,  §  166. 

Mortgage,  §§  167,  182. 

Subordinate,  §  160. 
Life  tenants,  payments  to,  §§  148-154. 
Little's  table  of  multiples,  §361;  comment,  §356. 
Loans, 

On  collateral,  §§  185-188. 

On  bond  and  mortgage,  §§  162-184. 

Periodic  payment,  §319. 

Uneven,  §  282. 
Logarithms,  Chs.  Ill,  XXIX,  XXX. 

Accuracy  of  results,  §§  50,  51. 

Application  of. 

To  amount  of  annuity,  §61. 

To  present  worth  of  annuity,  §  73, 

Bases  of,  §357. 

Characteristic,  §41. 

Division  by,  §  45. 

Factoring  method,  §§  343-346. 

Finding  numbers  from,  §§337-347. 


365 


366  INDEX 

Logarithms  (Continued) 
Forming,  §§  348-357. 

Multiplying  up,  §§  353-356. 
In  connection  with  effective  rates,  §  95. 
In  connection  with  valuation  of  bonds,  §  120. 
Mantissa,  §§41,  42. 
•      Multiple  method,  §347. 
Multiplication  by,  §  44. 
Powers,  finding,  §  46. 
Problems,  §§229-231. 
Roots,  finding,  §47. 
Rules  for  use  of,  §§  40,  44-47. 
Tables, 

Gray  and  Steinhauser,  §  341. 

Standard,  §339. 

To  fifteen  places,  §359;  comment,  §51. 

To  four  places,  §  43. 

To  twelve  places,  §358;  comment,  §§342-357. 

United  States  Coast  Survey,  §  340. 
Two  parts  of,  §  41. 
Use  of,  §§  36,  37. 

In  compound  interest  computations,  §49. 

In  present  worth  computations,  §  52. 
Loose-leaf  records  for  mortgages,  §  178. 


Mantissa,  §§41,  42. 

Market  value  of  bonds,  §  107. 

Month  as  time  unit,  §  21. 

Mortgages,  §§  163-184. 

Multiples,  Little's  table  of,  §361;  comment,  §356. 

Multiplication, 

By  logarithms,  §  44. 

Contracted,  §228. 

Tabular,  §248. 


N 


Net  income,  problems,  §  293. 
Nominal  rates,  §  91. 

Problems,  §§  236-240. 
Notes,  §§217-220. 
Notices  of  interest,  §  181. 


INDEX  367 


O 

Optional  redemption,  bonds  with,  §  147. 

Problems,  §§  287-294. 
Ordinary  annuities,  §§  74-76. 


Payments,  periodic,  §§  319-336. 
Periods, 

Annuity  and  interest  compared,  §  81. 

Interest,  §§  20-26,  364,  374. 
Perpetuities,  §§  79,  80. 
Powers,  §§  27,  38. 

Finding,  by  logarithms,  §  46. 
Premiums  on  bonds. 

Accounts,  §§  207,  209,  210,  214. 

Amortization  of,  §  328. 

Analysis  of,  §  317. 

Deferred,  §315. 

Immediate,  §314. 

Valuation  of,  §§  329,  330. 
Prepaid  annuities,  §§  75,  l(i. 
Present  worth,  §§29-31,  261. 

By  differences,  §  259. 

By  division,  §  260. 

Formulas  for,  §  67. 

Logarithmic  computation  of,  §  52. 

Of  annuities  due,  §§  75,  Id. 

Of  bonds,  §§101,  103,  106. 

Of  coupons,  §§  113,  296. 

Of  deferred  annuities,  §§  11 ,  78. 

Of  ordinary  annuity,  §§  62-73,  304. 

Of  perpetuities,  §§79,  80. 

Of  principal,  §  112. 

Proof  of  by  reciprocal,  §  227. 

Relation  to  amortization,  §  71. 

Short  method  for,  §§  64-67. 
Present  worths, 

Of  $1,  tables,  §§63,  379;  comment,  §369. 

Of  annuity  of  $1,  table,  §  381 ;  comment,  §  371. 

Of  interest-diflerence,  §271. 
Principal,  §  12. 

Account,  §§  164,  169,  179,  186-188,  202. 

Change  in,  §  290. 

Definition  of,  §  10. 


368  INDEX 

Problems, 

Annuities,  §§  232,  233. 

Bonds  at  annual  and  successive  rates,  §§295-318. 

Bonds  with  optional  redemption,  §§287-294. 

Interest, 

Compound,  §§  225,  226. 

Simple,  §§  222-224. 
Logarithms,  §§229-231. 
Nominal  and  effective  rates,  §§  236-240. 
Rent  of  annuity,  §§  234,  235. 
Serial  bonds,  §§  274-286. 
Sinking  funds,  §§  234,  235. 

Valuation  of  bonds,  §§243-318.    (See  also  "Bond  Problems.") 
Punctual  interest,  §§  14^  17. 

Q 

Quarter-year,  legal  definition  of,  §  22. 


Rates, 

Annual  and  other,  §§295-318. 

Conversions  of,  §  92. 

Coupon  (cash),  §§  100,  109. 

Definition  of,  §  10. 

Dual,  §§  326-333. 

Effective,  §§91-95,  100. 
Problems,  §§  236-240. 

Equivalent  (annual,  semi-annual  and  quarterly),  §§  241,  242. 

Income,  on  bonds,  §§  109,  134,  136,  137,  138-140,  263-269,  287,  291. 

In  interest  tables,  §  364. 

Inter-rates,  §§  275,  375,  Z76. 

Logarithmic  method,  §  95. 

Nominal,  §§91,  97. 

Problems,  §§  236-240. 

Trial,  §§  264-269. 
Ratios  of  increase,  §  17. 

Logarithms  of,  fifteen-place  table,  §359;  comment,  §51. 
Real  estate  mortgages,  §§  163-184. 
Receipts  and  notices,  interest,  §  181. 
Reciprocals, 

Amount  and  present  worth  as,  §  227. 

Meaning  of,  §  52. 

Table  of,  §  383;  comment,  §  Z77. 


INDEX  36^ 

Redemption  of  bonds,  §§  146,  147. 

Problems,  §§  287-294. 
Register, 

Collateral,  §§  186,-188. 

Interest,  §§  180,  183,  192,  193. 
Rent, 

Definition  of,  §6. 

Of  annuity,  §§  83-86,  Z73. 

Of  deferred  payments,  §86. 
Problems,  §§  234,  235. 
Repayment  and  reinvestment,  §§319-336. 
Replacement,  §§  148,  322,  323. 
Residues  on  bonds,  eliminating,  §§  138-140,  280. 
Reussner's  tables,  §  273. 
Revenue,  forms  of,  §  5. 
Roots,  §  38. 

Finding,  by  logarithms,  §  47. 
Rules  and  formulas,  §§  35,  157-159. 


Security,  collateral,  §§  185-188. 
Serial  bonds,  §  145. 

Problems,  §§  274-286. 
Series,  §§  30-32. 

Amortization,  §§  72,  249. 

Of  annuity  amounts,  §56. 
Short  terminal  bonds,  §§141,  142. 

Problems,  §§255,  256. 
Simple  (single)  interest,  §§  13,  16,  19. 

Problems,  §§  222-224. 
Single  discount,  %%ZZ,  216-218. 
Sinking  funds,  §§  87-90. 

Problems,  §§  234,  235. 

Relation  to  amortization,  §  90. 

Table,  §382;  comment,  §372. 
Square  roots,  table  of,  §383;  comment,  §377. 
Steinhauser's  tables  of  logarithms,  §  341. 
Subordinate  ledgers,  §  160. 
Sub-reciprocals,  table  of,  §  360. 
Successive  method,  §  277. 

Problems,  §244. 
Successive  rates,  bonds  at,  §§313-318. 
Symbols,  explanation  of,  page  xviii. 


370  INDEX 

T 

Tables, 

Amounts  of  $1,  §378;  comment,  §367. 

Amounts  of  annuity  of  $1,  §380;  comment,  §370. 

Bonds,  §  155;  comment,  §  156. 

Differences,  §  276. 

Four-place,  §  43. 

Interest  ratios  to  fifteen  places,  §359;  comment,  §51. 

Multiples,  §361;  comment,  §356. 

Present  worths  of  $1,  §§63,  379;  comment,  §369. 

Present  worths  of  annuity  of  $1,  §381;  comment,  §371. 

Reciprocals,  §383;  comment,  §377. 

Reussner's,  comment,  §  273. 

Sinking  funds,  §382;  comment,  §372. 

Square  roots,  §383;  comment,  ^377. 

Sub-reciprocals,  §  360. 

Twelve-place  logarithmic  factors,  §358;  comment,  §§342-357. 
Tabular  multiplication,  §  248. 
Time  units,  §§  10,  23. 

Day,  §§  20,  24. 

Half  and  quarter  years,  §  22. 

Month,  §21. 
Trust  funds,  §§  148-154. 

Cullen  decision,  §§  152-154. 
Twelve-place  logarithms,  tables,  §358;  comment,  §§342-357. 


U 

United  States  Coast  Survey  tables,  §  340. 


Valuation  of  bonds,  Chs.  X,  XI. 

First  method  (two  operations),  §§111-114. 

Interpolation  method,  §§  131,  132. 

Multiplication  method,  §§  133,  134. 

Periodic,  §§  130-133. 

Problems,  §§243-318. 

Rule  for,  §§305,  316.     ■ 

Schedules,  §§  114,  122,  126,  127,  130. 

Second  method  (one  operation),  §§115-120. 
Valuation  of  premiums,  §§  329.  330. 


INDEX  2^j 


Values, 

Discounted,  §§  215-221. 
Of  bonds,  §  127. 

Book,  §§  123,  125. 

Found  by  discounting,  §§  143,  144. 

Initial  book,  §  247. 

Intermediate,  §§  105,  376. 

Market,  §  107. 

Various,  §§  104-108. 
Varying  annuities,  §  82.  / 

y  ^ 

Year,  legal,  §§23,  24. 


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